random walk in random and non-random environments - by pal revesz

RANDOM WALK

IN RANDOM AND

NON-RANDOM ENVIRONMENTS

Pal ReveszTechnical University of Vienna, Austria

Technical University of Budapest, Hungary

World ScientificSingapore • New Jersey • London • Hong Kong

NON-ACTIVATEDVERSIONwww.avs4you.com

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World Scientific Publishing Co. Pte. Ltd.,

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USA office: 687 Hartwell Street, Teaneck, NJ 07666

UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Library of Congress Cataloging-in-Publication Data

Revesz, Pal.

Random Walk in random and non-random environments/Pal Revesz.

p. cm.

Includes bibliographical references (P. ) and indexes.

ISBN 9810202377

1. Random walks (Mathematics) I. Title.

QA274.73.R48 1990

519.2'82-dc20 90-37709

CIP

Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproducedin any form or by any means, electronic or mechanical, including photo-photocopying, recording or any information storage and retrieval system now

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Preface

"I did not know that it was so dangerous to drink a beer with you. You write

a book with those you drink a beer with," said Professor Willem Van Zwet,referring to the preface of the book Csorgo and I wrote A981) where it was told

that the idea of that book was born in an inn in London over a beer. In spiteof this danger Willem was brave enough to invite me to Leiden in 1984 for a

semester and to drink quite a few beers with me there. In fact I gave a seminar

in Leiden, and the handout of that seminar can be considered as the very first

version of this book. I am indebted to Willem and to the Department of Leiden

for a very pleasant time and a number of useful discussions.

I wrote this book in 1987-89 in Vienna (Technical University) partly sup-

supported by Fonds zur Forderung der Wissenschaftlichen Forschung, Project Nr.

P6076. During these years I had very strong contact with the Mathematical

Institute of Budapest. I am especially indebted to Professors E. Csaki and A.

Foldes for long conversations which have a great influence on the subject of this

book. The reader will meet quite often with the name of P. Erdos, but his role in

this book is even greater. Especially most results of Part II are fully or partly due

to him, but he had a significant influence even on those results that appearedunder my name only.

Last but not least, I have to mention the name of M. Csorgo, with whom I

wrote about 30 joint papers in the last 15 years, some of them strongly connected

with the subject of this book.

Vienna, 1989. P. Revesz

Technical University of Vienna

Wiedner Hauptstrasse 8-10/107A-1040 Vienna

Austria


Contents

Preface v

Introduction xiii

I. SIMPLE SYMMETRIC RANDOM WALK IN Zl

Notations and abbreviations 3

1 Introduction of Part I 9

1.1 Random walk 9

1.2 Dyadic expansion 10

1.3 Rademacher functions 11

1.4 Coin tossing 11

1.5 The language of the probabilist 11

2 Distributions 13

2.1 Exact distributions 13

2.2 Limit distributions 19

3 Recurrence and the Zero-One Law 23

3.1 Recurrence 23

3.2 The Zero-One Law 25

4 From the Strong Law of Large Numbers to the Law of

Iterated Logarithm 27

4.1 Borel - Cantelli Lemma and Markov inequality 27

4.2 The strong law of large numbers 28

4.3 Between the Strong Law of Large Numbers and the Law of

Iterated Logarithm 30

4.4 The LIL of Khinchine 31

vii


viii CONTENTS

5 Levy Classes 33

5.1 Definitions 33

5.2 EFKP LIL 35

5.3 The laws of Chung and Hirsch 39

5.4 When will Sn be very big? 39

5.5 A theorem of Csaki 41

6 Wiener process and Invariance Principle 47

6.1 Two lemmas 47

6.2 Definition of the Wiener process 48

6.3 Invariance Principle 52

7 Increments 55

7.1 Long head-runs 55

7.2 The increments of a Wiener process 63

7.3 The increments of Ss 73

8 Strassen type theorems 79

8.1 The theorem of Strassen 79

8.2 Strassen theorems for increments 86

8.3 The rate of convergence in Strassen's theorems 88

8.4 A theorem of Wichura 90

9 Distribution of the local time 93

9.1 Exact distributions 93

9.2 Limit distributions 99

9.3 Definition and distribution of the local time of a Wiener process 100

10 Local time and Invariance Principle 105

10.1 An Invariance Principle 105

10.2 A theorem of Levy 107

11 Strong theorems of the local time 113

11.1 Strong theorems for ?{x,n) and f(n) 113

11.2 Increments of ri(x,t) 115

11.3 Increments of ?{x,n) 118

11.4 Strassen type theorems 120

11.5 Stability 122

11.6 Favourite points . 129

11.7 Rarely visited points 132


CONTENTS ix

12 An embedding theorem 133

12.1 On the Wiener sheet 133

12.2 The theorem .134

12.3 Applications 137

13 Excursions 141

13.1 On the distribution of the zeros of a random walk 141

13.2 Local time and the number of long excursions

(Mesure du voisinage) 147

13.3 The local time of high excursions 152

13.4 How many times can a random walk reach its maximum? .... 157

14 A few further results 161

14.1 On the location of the maximum of a random walk 161

14.2 On the location of the last zero 165

14.3 The Ornstein - Uhlenbeck process and a theorem of

Darling and Erdos 169

14.4 A discrete version of the Ito formula 173

15 Summary of Part I 177

II. SIMPLE SYMMETRIC RANDOM WALK IN Zd

Notations 181

16 Recurrence theorem 183

17 Wiener process and Invariance Principle 189

18 The Law of Iterated Logarithm 193

19 Local time 197

19.1 f@, n) in Z2 197

19.2 f (n) in Zd 204

19.3 A few further results 206

20 The range 207

20.1 The range of Sn 207

20.2 Wiener sausage 210


x CONTENTS

21 Selfcrossing 213

22 Large covered balls 217

22.1 Completely covered discs in Z2 217

22.2 Discs covered with positive density 233

22.3 Completely covered balls in Zd 241

22.4 Once more on Z2 248

23 Speed of escape 249

24 A few further problems 255

24.1 On the Dirichlet problem 255

24.2 DLA model 258

24.3 Percolation 260

III. RANDOM WALK IN RANDOM ENVIRONMENT

Notations 263

25 Introduction 265

26 In the first six days 269

27 After the sixth day 273

27.1 The recurrence theorem of Solomon 273

27.2 Guess how far the particle is going away in an RE 275

27.3 A prediction of the Lord 277

27.4 A prediction of the physicist 287

28 What can a physicist say about the local time f@,n)? 291

28.1 Two further lemmas on the environment 291

28.2 On the local time f@, n) 292

29 On the favourite value of the RWIRE 297

30 A few further problems 305

30.1 Two theorems of Golosov 305

30.2 Non-nearest-neighbour random walk 307

30.3 RWIRE in Zd 308

30.4 Non-independent environments 310


CONTENTS xi

30.5 Random walk in random scenery 310

30.6 Reinforced random walk 311

References 313

Author Index 327

Subject Index 331


Introduction

The first examinee is saying: Sir, I did not have time enough to study everythingbut I learned very carefully the first chapter of your handout.

Very good -

says the professor -

you will be a great specialist. You know what

a specialist is. A specialist knows more and more about less and less. Finally he

knows everything about nothing.The second examinee is saying: Sir, I did not have enough time but I read your

handout without taking care of the details. Very good - answers the professor -

you will be a great polymath. You know what a polymath is. A polymath knows

less and less about more and more. Finally he knows nothing about everything.

Recalling this old joke and realizing that the biggest part of this book is

devoted to the study of the properties of the simple symmetric random walk (orequivalently, coin tossing) the reader might say that this is a book for specialistswritten by a specialist. The most trivial plea of the author is to say that this

book does not tell everything about coin tossing and even the author does not

know everything about it. Seriously speaking I wish to explain my reasons for

writing such a book.

You know that the first probabilists (Bernoulli, Pascal, etc.) investigated the

properties of coin tossing sequences and other simple games only. Later on the

progress of the probability theory went into two different directions:

(i) to find newer and deeper properties of the coin tossing sequence,

(ii) to generalize the results known for a coin tossing sequence to more com-

complicated sequences or processes.

Nowadays the second direction is much more popular than the first one. In

spite of this fact this book mostly follows direction (i). I hope that

(a) using the advantage of the simple situation coming from concentrating on

coin tossing sequences, the reader becomes familiar with the problems, results

and partly the methods of proof of probability theory, especially those of the

limit theorems, without suffering too much from technical tools and difficulties,

(b) since the random walk (especially in Zd) is the simplest mathematical

model of the Brownian motion, the reader can find a simple way to the problems

xiii


xiv INTRODUCTION

(at least to the classical problems) of statistical physics,(c) since it is nearly impossible to give a more or less complete picture of the

properties of the random walk without studying the analogous properties of the

Wiener process, the reader can find a simple way to the study of the stochastic

processes and should learn that it is impossible to go deeply in direction (i)without going a bit in direction (ii),

(d) any reader having any degree in math can understand the book, and

reading the book can get an overall picture about random phenomena, and the

readers having some knowledge in probability can get a better overview of the

recent problems and results of this part of the probability theory,

(e) some parts of this book can be used in any introductory or advanced

probability course.

The main aim of this book is to collect and compare the results -mostly strongtheorems - which describe the properties of a simple symmetric random walk.

The proofs are not always presented. In some cases more proofs are given, in

some cases none. The proofs are omitted when they can be obtained by routine

methods and when they are too long and too technical. In both cases the reader

can find the exact reference to the location of the (or of a) proof.


"Four legs good, two legs better."

A modified version of f.he

Animal Farm's Constitution.

"Two logs good, p logs better."

The original Constitution

of mathematicians.

I. SIMPLE SYMMETRIC

RANDOM WALK IN lNON-ACTIVATEDVERSION

www.avs4you.com

Notations and abbreviations

Notations

General notations

1. Xi,X2,... is a sequence of independent, identically distributed random

variables with

2. So = O,Sn = S{n) = Xl + X2 + --- + Xn (n = 1,2,...).{Sn} is the (simple symmetric) random walk.

3. M+ = M+(n) = maxSk,n V '0<k<n

'

M~ = M~(n) = — min S*,n V '0<k<n

*'

Mn = M(n) = max |5jt| = max(M^",M~),0<fc<n

4. {W(t);t > 0} is a Wiener process (cf. Section 6.1).

5. m+{t) = supW(s),0<»<t

m-{t) = - inf W(s),0<s<t

V '

m(t) = sup \W(s)\ = max(m+{t),m-{t)) {t > 0),0<»<t

m*(t) = m+(t)+m-{t),= m+{t)-W{t).

6. 6n = 6(n) = Bnloglogn)-1/2,In = l{n, a) = 12a Hog - + log log n) )

7. [x] is the largest integer less than or equal to x.

I. SIMPLE SYMMETRIC RANDOM WALK IN Zl

8. f(n) > g(n) «-> g(n) = o(f(n)) «-> Jun-—¦ = oo.

9. g(n) = O(f(n)) <-+ 0 < Iiminf44 < Iimsup44 <

10. /(n) « g(n) <-> lim —)—r- = 1.

11. Sometimes we use the notation f(n) ~ g(n) without any exact mathemati-

mathematical meaning, just saying that f(n) and g(n) are close to each other in some

sense.

12. $(z) =.— / e~u /2du is the standard normal distribution function.

13. N eN{m,a) ^P{a~l(N - m) <x} = $(z).

14. #{...} =| {...} | is the cardinality of the set in the bracket.

15. Rd resp. Zd is the d-dimensional Euclidean space resp. its integer grid.

16. B = Bd is the set of Borel-measurable sets of Rd.

17. A() is the Lebesgue measure on Rd.

18. logp (p = 1,2,...) is p-th iterated of log and lg resp. lgp is the logarithmresp. p-th iterated of the logarithm of base 2.

19. Let {Un} and {Vn} be two sequences of random variables.

{Un,n= 1,2,...} = {Vnn = 1,2,...} if the finite dimensional distributions

of {Un} are equal to the corresponding finite dimensional distributions of

Notations to the increments

1. h{n,a) = max (Sk+a - Sk),0<k<n-a

2. h{n,a) = QnnxJSk+a- Sk\,

3. h(n, a) = max

4. lAn,a) = max max|Sjt+,- — Sid,

NOTATIONS AND ABBREVIATIONS 5

5. h{n,a) = min max I S*+,-— S* |,0<k<n~a0<j<a

6. J1{t,a)= sup {W{s + a) -W{s)),0<S<t-a

7.J2{t,a)= sup \W(s + a)-W{s)\,0<s<t-a

8. J3(t,a)= sup sup (W(s+u)- W(s)),0<t<t-a0<u<a

9. J4{t, a) = sup sup \W(s + u) - W(s)\,0<3<t-a0<u<a

10. JM,a)= inf sup \W(s + u) - W(s)\,0<«<t-a0<u<a

11. Zn < n is the largest integer for which h(n, Zn) = Zn, i.e. Zn is the lengthof the longest run of pure heads in n Bernoulli trials.

Notations to the Strassen-type theorems

1. Sn(x) = bn Unz] +(x- I~H X[nx]+1\ @ < x < 1),

2. iut(x) = btW(tx) @<x<l;t>0),

3. C@,1) is the set of continuous functions defined on the interval [0,1],

4. S@,1) is the Strassen's class, containing those functions /(•) ? C@,1) for

which /@) = 0 and ^{f'{x))Hx < 1.

Notations to the local time

1. ?(x, n) = #{k : 0 < k < n, Sk = x} (x = 0, ±1, ±2,...; n = 1,2,...) is

the local time of the random walk {Sk}. For any A C Z1 we define the

occupation time E(A, n) = Ezex f (x>n)-

2. 77B:,*) (-cx> < x < +oo;t > 0) is the local time of W(-) (cf. Section 9.3).

3. H(A,t) = \{s : 0 < 5 < t,W(s) 6 A} (A c /21 is a Borel set, t > 0) is the

occupation time of W(-) (cf. Section 9.3).

4. Consider those values of k for which S* = 0. Let these values in increasingorder be 0 = p0 < pl < p2 < ..., i.e. pl = min{A; : k > 0, Sk = 0},p2 =

mm{k : k > Pl, Sk = 0},..., pn = m\n{k : k > pn-USk = 0},....

I. SIMPLE SYMMETRIC RANDOM WALK IN Z1

5. Similarly for any x = 0, ±1,±2,... consider those values of k for which

Sk = x. Let these values in increasing order be 0 < Pi(x) < p2(x) < ...

i.e. Pi(x) = min{A; : k > 0, Sk = x},p2(x) = min{A; : k > pi(x),Sk =

x}-> • • • ,Pn{x) = min{A; : k > pn-i(x),Sk = x] ... Clearly p,@) = /v In case

of a Wiener process define p*u = inf{t : t > O,ij(O,t) > u}.

6. ^(n) = max, ?(x,n).

7. ri(t) =supxV(x;t).

8. The random sequences

Si,... ,SPi}, E2 = \SP1,

are called the first, second, ... excursions (away from 0) of the random

walk {Sk}.

9. The random sequences

Ei(x) =

E2(x) =

are called the first, second, ... excursions away from x of the random walk

is*}.

10. For any t > 0 let a(t) = sup{r : r < t,W{r) = 0} and f3(t) = inf{r :

t > t,W(r) = 0}. Then the path {Wt(s);a{t) < s < (l(t)} is called an

excursion of W(-).

11. fn is the number of those terms of Si, 52,..., Sn which are positive or which

are equal to 0 but the preceding term of which is positive.

12. 0(n) = #{k : 1 < k < n, Sk-iSk+i < 0} is the number of crossings.

13. R(n) = max{A; : k > 1 for which there exists a0<j<n-k such that

f@).7 + k) = f@, j)} is the length of the longest zero-free interval.

14. r(t) = sup{s : 5 > 0 for which there exists a0(t) = sup{5 : 0 < 5 < t, W{s) = 0}.

NOTATIONS AND ABBREVIATIONS 7

17. R(n) = max{A: : k > 1 for which there exists a.0<j<n-k such that

M+(j + k) = M+(j)} is the length of the longest flat interval of M? up to

n.

18. f[i) = sup{s : 5 > 0 for which there exists aO 1 for which there exists a 0 < j < n - k such that

20. r*(t) = sup{s : 5 > 0 for which there exists a0< u<f-s such that

m(u + s) = m(u)}.

21. (i(n) is the location of the maximum of the absolute value of a random

walk {Sk} up to n, i.e. fj.(n) is defined by S(fx(n)) = M(n) and fx(n) < n.

If there are more integers satisfying the above conditions then the smallest

one will be considered as /x(n).

22. M{t) = inf{s : 0 < 5 < t for which W(s) = m(t)}.

23. fi+{n) = [nf{k : 0 < k < n for which S{k) = M+{n)}.

24. M+{t) = inf{5 : 0 < 5 < t for which W{s) = m+{t)}.

25. x(n) is the number of those places where the maximum of the random walk

So, Si,..., Sn is reached, i.e. x(n) is the largest positive integer for which

there exists a sequence of integers 0 < kx < k2 < ... < kx(n) < n such that

S{kx) = S(k2) = • - • = S(kx{n)) = M+(n).

Abbreviations

1. r.v. = random variable,

2. i.i.d.r.v.'s = independent, identically distributed r.v.'s,

3. LIL = law of iterated logarithm,

4. UUC, ULC, LUC, LLC, AD, QAD (cf. Section 5.1),

5. i.o. = infinitely often,

6. a.s. = almost surely.

Chapter 1

Introduction of Part I

The problems and results of the theory of simple symmetric random walk in Z1

can be presented using different languages. The physicist will talk about random

walk or Brownian motion on the line. (We use the expression "Brownian motion"

in this book only in a non-well-defined physical sense and we will say that the

simple symmetric random walk or the Wiener process are mathematical models of

the Brownian motion.) The number theorist will talk about dyadic expansions of

the elements of [0,1]. The people interested in orthogonal series like to formulate

the results in the language of Rademacher functions. The gambler will talk about

coin tossing and his gain. And a probabilist will consider independent, identicallydistributed random variables and the partial sums of those.

Mathematically speaking all of these formulations are equivalent. In order

to explain the grammar of these languages in this Introduction we present a few

of our notations and problems using the different languages. However, later on

mostly the "language of the physicist and that of the probabilist" will be used.

1.1 Random walk

Consider a particle making a random walk (Brownian motion) on the real line.

Suppose that the particle starts from x = 0 and moves one unit to the left with

probability 1/2 and one unit to the right with probability 1/2 during one time

unit. In the next step it moves one step to the left or to the right with equalprobabilities independently from its location after the first step. Continuing this

procedure we obtain a random walk that is the simplest mathematical model of

the linear Brownian motion.

Let Sn be the location of the particle after n steps or in time n. This model


10 CHAPTER 1

clearly implies that

P{Sn+i = tn+i | Sn = in, 5n_i = in_i,..., Si = »i, So = t0 = 0}

= P{Sn+1 = in+1 | Sn = in} = 1/2 A.1)

where i0 = 05 *i, »2) • • • ¦> *n, *n+i is a sequence of integers with |t'i — »o| = |»2 — »i| =

...= |tn+1 — tn| = 1. It is also natural to ask: how far does the particle go away

(resp. going away to the right or to the left) during the first n steps. It means

that we consider

Mn = max \Sk\ resp. M+ = max Sk or M~ = — min S*.

1.2 Dyadic expansion

Let x be any real number in the interval [0,1] and consider its dyadic expansion

x = 0,1

where e< = e,(x) (t = 1,2,...) is equal to 0 or 1. In fact

et = [2'z] (mod 2).

Observe that

\{x : *,-,(*) = 6ush(x) = S2,.. .,ejn(x) = Sn} = 2~n A.2)

where 1 < j\ < j2 < ... < jn;n = 1,2,...; 6i, 62,..., 6n is an arbitrary sequence

of 0's and +l's and A is the Lebesgue measure. Let So = ^o(^) = 0 and Sn =

Sn(x) = n - 2 ?r=i ei(x) (n = 1,2,...). Then A.2) implies

A{x : Sn+1 = «n+1, Sn = in,..., Sl = »!, So = to} = 2~(n+1) A.3)

where t'o = 0, iu i2,..., in+1 is a sequence of integers with |t'i — t'o| = |z*2 — »i| =

...= |tn+i — tn| = 1. Clearly A.3) is equivalent to A.1). Hence any theorem

proved for a random walk can be translated to a theorem on dyadic expansion.A number theorist is interested in the frequency Nn(x) = Z)"=1e,(x) of the

ones among the first n digits of x 6 [0,1]. Since Nn(x) = |(n — Sn(x)) anytheorem formulated for Sn implies a corresponding theorem for Nn(x).

INTRODUCTION OF PART I 11

1.3 Rademacher functions

In the theory of orthogonal series the following sequence of functions is well-

known. Let

r(x)-i l if *e [0,1/2),rilxj-\-i if x e [i/2,i],

, f 1 if x€[0,l/4)U[l/2,3/4),r2lXj-\-l if x€[l/4,l/2)U[3/4,l],

, f 1 if x€ [0,1/8) U[l/4,3/8) U[l/2,5/8) U[3/4,7/8),r3lXj-\-l if x€[l/8,l/4)U[3/8,l/2)U[5/8,3/4)U[7/8,l],...

An equivalent definition, by dyadic expansion, is

rn(x) = l-2en(x).

The functions ri(x),r2(x),... are called Rademacher functions. It is a se-

sequence of orthonormed functions, i.e.

Observe that

\{x : rh(x) = 61,rh(x) = 62,.. .,rjn(x) = 6n} = 2~n A.4)

where 1 < j\ < j2 < ... < jn, n = 1,2,...; Si, 62,..., Sn is an arbitrary sequence

of -fl's and —l's and A is the Lebesgue measure. Putting So = S0(x) = 0 and

Sn = Sn{x) = E,n=i »\-(x)(n = 1,2,...) we obtain A.3).

1.4 Coin tossing

Two gamblers (A and B) are tossing a coin. A wins one dollar if the tossingresults in a head and B wins one dollar if the result is tail. Let Sn be the amount

gained by A (in dollars) after n tossings. (Clearly Sn can be negative and So = 0

by definition.) Then 5^ satisfies A.1) if the game is fair, i.e. the coin is regular.

1.5 The language of the probabilistLet Xi, X2,... be a sequence of i.i.d.r.v.'s with

P{Xt = 1} = P{Xt = -1} = 1/2 (» = 1,2,...),

12 CHAPTER I

i.e.

P{Xn = *lt Xh = *„..., XJn = U = 2"" A.5)

where 1 < jx < j2 < ... < jn;n = 1,2,... and 6i,62,...,6n is an arbitrarysequence of -f l's and — l's. Let

n

50 = 0 and Sn = ? *k (n = l,2,...).

Then A.5) implies that {Sn} is a Markov chain, i.e.

, Sn-l = »n-l) • ¦ • ) ^1 = *1> ^0 = *0 = 0} =

= *n+i I Sn = in} = 1/2 A.6)

where t'o = 0, i\, i2,..., tn, tn+1 is a sequence of integers with |z'i — t'o| = |»2 — t'i| =

...= |»n+1 -»n| = 1.

Chapter 2

Distributions

2.1 Exact distributions

A trivial combinatorial argument gives

THEOREM 2.1

where k = —n, —n + 1,..., n; n = 1,2,...,

= 2* + l}=Bn+IV2"-1 B.2)

where k = —n — 1,—n,... ,n; n = 1,2, —

B.1) and B.2) together give

P{Sn = A:} =

= n (mod2)'

0 otherwise

where k = —n, —n + 1,..., n; n = 1,2, —

Further, for any n = 1,2,..., t 6 -R1 we

B.3)

ESn = 0, E5^ = n, Eexp(*Sn) = (-^—

1 . B.4)

The following inequality (Bernstein inequality) can also be obtained by ele-

elementary methods:

13


14 CHAPTER 2

THEOREM 2.2 (cf. e.g. Renyi, 1970/B, p. 387).

2ne2Sn

n>e\ <2expf-

for any n = 1,2,... and 0 < e < 1/4.

For later reference we present also a slightly more general form of the Bern-

Bernstein inequality.

THEOREM 2.3 (cf. Renyi, 1970/B, p. 387). Let X{,X;,... be a sequence ofi.i.d.r.v.'s with

Then for any 0 < e < pq we have

>e\ <2exp

2pq 11 +

where S; = X\ + X\ + • • • + X*n and q = 1 - p.

THEOREM 2.4 (cf. e.g. Renyi, 1970/A, p. 233).

2pql

n+

= k}= [\n-k B.5)

and for any t ? R1

2n

= Eexp(*M2+J =+J = 2~kt

Jk=O

Proof 1 of B.5). (Renyi, 1970/A, p. 233). Let

F+Mn = max } Xj.

J=2

Then

B.6)

>fc= P{X1 = l,M^=A:-l} + = *+¦!}

DISTRIBUTIONS 15

-(Pn,Jk-l +Pn,Jk+l) > 1).

Similarly for A: = 0

,= -1,M+ < 1} = i(p^ + pn>0).Pn+1,0 =

Since p10 = pu = 1/2 we get B.5) from B.7) and B.8) by induction.

Proof 2 of B.5). Clearly

P{M+ > A:} = P{5n > A:} + P{5n < k,M+ > k}( n

B.7)

B.8)

j=n (mod 2)

n—j^ 2

Let

Pi(A:) = min{/ : 5, = *},if

i if

i.e. Si for / > p\(k) is the reflection of Sj in the mirror y = k. (Hence the

method of this proof is called reflection principle.) Then

\n-jj=n (mod 2)

n / n

j=n (mod 2)

= 2"" E (n-j)+2-»j'=n (mod 2) j=n (mod 2)

= 2P{5n > A:} + P{5n = A:} = 2~n EJ=Jk

n

n-j

B.9)

which proves B.5).B.6) can be obtained by a direct calculation.

16 CHAPTER 2

THEOREM 2.5 For any integers a<O<b,a<b,a<u<bwe have

pn{a, b, u) = P{a < -M- < M+ <b,Sn = u)

= f) qn{u + 2k{b - a)) - ?) qnBb - u + 2k(b - a)) B.10)Jk=-oo fc=-oo

where

0 otherwise

(j = -n,-n + l,...,n;n = 0,1,...)-

Proof. (Billingsley, 1968, p.78). In case n = 0

... { 1 if v = 0 and a2 + b2 > 0,Po(a,b,u) =

jQ otherwise?

and we obtain B.10) easily. Assume that B.10) holds for n — 1 and for any a, b, w

satisfying the conditions of the Theorem. Now we prove B.10) by induction.

Note that pn@,b, w) = pn(a,0,u) = 0 and the same is true for the righthand side

of B.10) (since the terms cancel because qn{j) = <7n(-./))• Hence we may assume

that a < 0 < b. But in this case a + 1 < 0 and b — 1 > 0. Hence by induction

B.10) holds with parameters n — 1, a + 1, b + 1, u and n — 1, a — 1, b — 1, u. We

obtain B.10) observing that

and

pn(a, 6, j/) = -pn-i(a - 1,6 - 1,i/ - 1) + -Pn-i(a + 1,6 + 1,i/ + 1).

THEOREM 2.6 For any integers a < 0 < b and a<u<u<bwe have

P{a < -M~ < M+ < b, u < Sn < u}

f a) < Sn<v + 2k{b-a)}k=-oo

~ E P{2b-v + 2k{b-a) < Sn<2b-u + 2k{b-a)}, B.11)Jk=-oo

DISTRIBUTIONS 17

P{a < -M~ <M+<b}

= E P{a + 2k{b-a) <Sn<b + 2k{b-a)}k=-oo

- E ?{b + 2k{b-a) < Sn < 2b - a + 2k{b - a)} B.12)Jk=-oo

and

P{Mn < 6} = E P(D* " !N <Sn< D*Jfc=-oo

Sn< DA: + 3N}. B.13)Jk=-oo

B.11) is a simple consequence of B.10), B.12) follows from B.11) taking u =

a,v = b and B.13) follows from B.12) taking a = -b.

To evaluate the distribution of Ii(n,a) (i = 1,2,3,4,5) seems to be very

hard (cf. Notations to the Increments). However, we can get some information

about the distribution of Ji(n,a).

LEMMA 2.1 (Erdos - Revesz, 1976).

p{n+j,n) = P{Il{n + j,n) = n) = 3-^T (j = 0,1,2,..., n).

Clearly p(n + j, n) is the probability that a coin tossing sequence of lengthn + j contains a pure-head-run of length n.

Proof. Let

A = {I^n + j,n) = n} and Ak = {Sk+n - Sk = n}.

Then

A = Ao + A0Ai + A0AiA2 + • • • + A0Ai • ¦ • Aj^iAj= Ao + A0Ai + AxAi H h Aj-iAj.

Since P(A0) = 2~n and PiAoAi. ..M>i) = 2~n-1 for any t = 1,2,... ,j- 1 we

have the Lemma.

The next recursion can be obtained in a similar way.

18 CHAPTER 2

LEMMA 2.2 For any j = 1,2,... we have

Bn + j, n) = n} = pBn + j, n)

2n+l'

A - p(n + 2, n))^ + • • • + A - P(n + j- 1, n)) —

In case j < n we obtain

In some cases it is worthwhile to have a less exact but simpler formula. For

example, we have

LEMMA 2.3 (Deheuvels - Erdos - Grill - Revest, 1987).

{j + 2J-*-J - {j + 2J2�n~2 < P{h{n + j,n) = n) < {j + 2J"n� B.14)

for anyn = 1,2,... ;j = 1,2,....

The idea of the proof is the same as those of the above two lemmas. The

details are omitted.

The exact distribution of Zn (cf. Notations to the Increments) is also known,

namely:

THEOREM 2.7 (Szekely - Tusnady, 1979).

where

Remark 1. Csaki, Foldes and Komlos A987) worked out a very general method

to obtain inequalities like B.14) .Their method gives a somewhat weaker result

than B.14). However, their result is also strong enough to produce most of the

strong theorems given later (cf. Section 7.3).

DISTRIBUTIONS 19

2.2 Limit distributions

Utilizing the Stirling Formula

(where 0 < an < 1) and the results of Section 2.1, the following limit theorems

can be obtained.

THEOREM 2.8 (e.g. Renyi, 1970/A, p. 208). Assume that for some 0 < e <

1/2 the inequality en < k < A — e)n is satisfied. Then

(n\ . _2"«jr>

where K = k/n and d(K) = Klog2K + A - K) log2A - K). If we also assume

that \k-n/2\= o{n2lz) then

Especially

2nJ-2nThe next theorem is the so-called Central Limit Theorem.

THEOREM 2.9 (Gnedenko - Kolmogorov, 1954, §40).

<x}- *(z)| < 2m�/2.

A stronger version of Theorem 2.9 is the so-called Large Deviation Theorem:

THEOREM 2.10 (e.g. Feller, 1966, p. 517).

Pjn-1/2^ < -xn} F{n-l'*Sn > xn}hm —i—— = hm —*-^y—.—-n—oo $(-Xn) n—oo 1 — $(xn)

provided that 0 < xn = o(n1/6).

Theorem 2.10 can be generalized as follows:

20 CHAPTER 2

THEOREM 2.11 (e.g. Feller, 1966, p. 517). Let X{,X;,... be a sequence

of i.i.d.r.v. 's with

EX' = 0, E(X'J = I, Eexp(*X;) < oo

for all t in some interval \ t \< to. Then

P{n-^S^ < -xn} F{n-V*S; > xn}lim —

r = lim ——r = 1

$() 1 $()n-oo

provided that 0 < xn = o{n1/6) where S^ = X{ + X\ + ¦ • • + X*n.

THEOREM 2.12 (e.g. Renyi, 1970/A, p. 234).

lim F{n-1'2M+ < x} = P{\N\ < x} = 2S(z) - 1n—»oo

uniformly in x G R1 where N 6 iV@, l). Further,

lim E(n�/2Mn+) = B/ttI/2.

THEOREM 2.13

lim P{n-^2Mn <x} = G{x) = H{x)n—'oo

uniformly in x ? R1. Further,

v P{n/Mn > xn} Fjn^Mn < x}lim 7-7—r

= lim ——rr — 1n-00 1 - G(xn) n-°° H{1)

where

provided that 0 < xn = o{nxl*). Consequently for any e > 0

?{n-l'2Mn > xn} > A - e){l - G{xn))

>2A-

DISTRIBUTIONS 21

> xn} < A + e)(l - G(xn))

4A -e) f / 7T2 2\ 1 / 9tt2 2\1 . .

> A—I ^exp ^-T,iJ-

rxp (—i-^JJ B-18)

if 0 < xn = ©(n1/6) and n is 6«*flr enough.

Remark 1. As we claimed G{x) = #(z) however in Theorem 2.13 the asymp-

asymptotic distribution in the form of G(-) is proposed to be used when x is big. When

x is small H(-) is more adequate.Finally we present the limit distribution of Zn.

THEOREM 2.14 (Foldes, 1975, Goncharov, 1944). For any positive integer k

we have

P{Zn - [lg N}<k}= exp(-2-(fc+1)-^Ar>) + o(l)

where {lg iV} = lg iV - [lg N].

Remark 2. As we have mentioned earlier the above Theorems can be provedusing the analogous exact theorems and the Stirling formula. Indeed this method

(at least theoretically) is always applicable, it often requires very hard work.

Hence sometimes it is more convenient to use characteristic functions or other

analytic methods.

Chapter 3

Recurrence and the Zero-One Law

3.1 Recurrence

One of the most classical strong theorems on random walk claims that the particlereturns to the origin infinitely often with probability 1. That is

RECURRENCE THEOREM (Polya, 1921).

P(Sn = 0 i.o.) = 1.

We present three proofs of this theorem. The first one is based on the followinglemma:

LEMMA 3.1 Let 0 i we have

p@, t, k) = P{min{j : j > m, 5, = 0} < min{j : j > m, Sj = k} \ Sm = i}= k~1(k-i), C.1)

i.e. the probability that a particle starting from i hits 0 before k is k-1(k — i).

Proof. Clearly we have

p@,0,*) = l, p@,M)=0.

When the particle is located in t then it hits 0 before A: if

(i) either it goes to i — 1 (with probability 1/2) and from i — 1 goes to 0 before

A: (with probability p@,t - l,k)),

(ii) or it goes to i + 1 (with probability 1/2) and from » + 1 goes to 0 before A;

(with probability p@,t + l,k)).

23

24 CHAPTER 3

That is

p@, i, Ar) = ip(O,» - 1,*) + ip(O,» + 1,*)

(i = 1,2,..., A: — 1). Hence p@, i, k) is a linear function of i, being 1 in 0 and 0

in k, which implies C.1).

Proof 1 of the Recurrence Theorem. Assume that S\ = 1, say. By C.1)for any e > 0 there exists a positive integer no = no(e) such that p@, l,n) =

1 - 1/n > 1 - ?¦ if n > n0- Consequently the probability that the particle returns

to 0 is larger than 1 — e for any e > 0. Hence the particle returns to 0 with

probability 1 at least once. Having one return, the probability of a second return

is again 1. In turn it implies that the particle returns to 0 infinitely often with

probability 1.

Proof 2 of the Recurrence Theorem. Introduce the following notations

Po = 1,

P» = *{SU = 0} = 2~U^) (* = 1,2,...),

A2k = {S2k = 0, S2Jk-2 # 0,52Jk_4 # 0,..., S2 ? 0},q2k =

Jk=O

oo

(Note that q2k is the probability of the event that the first return of the particleto the origin occurs in the BA:)-th step but not before.)

Since p2k « (ttA;)�/2 (cf. Theorem 2.1) we have

limP(z) = oo.

Observe thatJk—1

{S2k = 0} = A2k + Y, A2k-2j{S2k = 0}

and

Y{A2k-2j{S2k = 0}} = q2k-2jp2j.


RECURRENCE AND THE ZERO-ONE LAW 25

Hence

@) Po = 1,

A) p2 = 92,

.(ii) P4 = 94 + 92P2,

(iii) p6 = 96 + 94P2 + 92P4,• • •

(k) P2Jk = 92Jfc + 92Jk-2P2 H H 92P2Jk-2,' ' '

Multiplying the Jfc-th equation by z2fc(|2| < 1) and summing up to infinity we

obtain

P(z)=P(z)Q(z)+l,i.e.

Q{z) = 1 " ^T and Jim Q(z) = 1.

Since Q(l) = YtkLitek = 1 is the probability that the particle returns to the

origin at least once we obtain the theorem.

3.2 The Zero-One Law

The above two proofs of the Recurrence Theorem are based on the fact that if

P{5jk = 0 at least for one n} = 1

then

P{5n = 0 i.o.} = 1.

Similarly one can see that if P{5n = 0 at least for one n) were less than 1 then

P{5n = 0 i.o.} would be equal to 0. Hence without any calculation one can

see that P{5n = 0 i.o.} is equal to 0 or 1. Consequently in order to prove the

Recurrence Theorem it is enough to prove that P{5n = 0 i.o.} > 0.

In the study of the behaviour of the infinite sequences of independent r.v.'s

we frequently realize that the probabilities of certain events can be only 0 or 1.

Roughly speaking we have : let Y\,Y2,... be a sequence of independent r.v.'s.

Then, if A is an event depending on Yn,Yn+\,... (but it is independent from

Y\, Yi,..., Yn-i) for every n, it follows that the probability of A equals either 0

or 1. More formally speaking we have

ZERO-ONE LAW (Kolmogorov, 1933). Let YuY2i... be independent r.v.'s.

Then if A ? Q is a set measurable on the sample space of Yn, Fn+i,... for every

n, it follows that

P{A) =0 or P(A) = 1.

26 CHAPTER 3

Example 1. Let YUY2,... be independent r.v.'s. Then ?,°li Y" converges a.s.

or diverges a.s.

Having the Zero-One Law we present a third proof of the Recurrence Theo-

Theorem. It is based on the following:

LEMMA 3.2 For any —oo < a < b < +oo we have

P{liminf Sn = a} = PlimsupSn = b} = 0.n—oo n-»oo

Proof is trivial.

Proof 3 of the Recurrence Theorem. Lemma 3.2, the Zero-One Law and

the fact that Sn is symmetrically distributed clearly imply that

P{liminf Sn = -oo} = P{limsup5n = oo} = 1 C.2)n—»oo

n—»oo

which in turn implies the Recurrence Theorem.

Note that C.2) is equivalent to the Recurrence Theorem. In fact the Recur-

Recurrence Theorem implies C.2) without having used the Zero-One Law.NON-ACTIVATEDVERSIONwww.avs4you.com

Chapter 4

From the Strong Law of Large Numbers

to the Law of Iterated Logarithm

4.1 Borel - Cantelli lemma and

Markov inequality

The proofs of almost all strong theorems are based on different forms of the Borel

- Cantelli lemma and those of the Markov inequality. Here we present the most

important versions.

BOREL - CANTELLI LEMMA 1 Let AUA2,... be a sequence of events forwhich Y%Li P(Ai) < oo. Then

P{limsup An} = P ( fl f>4 = P(^n i.o.) = 0,n-°° ln=l.=n )

i.e. with probability 1 only a finite number of the events An occur simultaneously.

BOREL - CANTELLI LEMMA 2 Let AUA2,... be a sequence of pairwiseindependent events for which Y^=\ P(-^n) = oo. Then

P{limsup An) = 1,n—»oo

i.e. with probability 1 an infinite number of the events An occur simultaneously.

BOREL - CANTELLI LEMMA 2*(Spitzer, 1964). Let AUA2,... be a se-

27

28 CHAPTER 4

quence of events for which

T P(/U = oo and liminf *=^=* -j-< C (C > 1).

Then

P{limsupAn} >C~l.n—»oo

MARKOV INEQUALITY Let X be a non-negative r.v. with EX < oo. Then

for any A > 0

P{X > XEX} < \.As a simple consequence of the Markov inequality we obtain

CHEBYSHEV INEQUALITY Let X be an r.v. with EX2 < oo. Then forany A > 0

P{|X - EX| > A(E(X - EXJI/2} = P{(X - EXJ > A2E(X - EXJ} < —.

Similarly we get

THEOREM 4.1 Let X be an r.v. with E(exp(*X)) < oo for some t > 0. Then

for any A > 0 we have

P{X > A} = P{exp(*X) > eAt} < 5^-.Borel - Cantelli lemmas 1 and 2 and Markov inequality can be found practi-

practically in any probability book (see e.g. Renyi, 1970/B).

4.2 The strong law of large numbers

THEOREM OF BOREL A909).

lim n^Sn = 0 a.s. D.1)

Remark 1. Applying this theorem for dyadic expansion we obtain

limn—oo n~1Nn(x) = 1/2 for almost all x ? [0,1]. In fact the original theorem

STRONG LAW AND LIL 29

of Borel was formulated in this form. Borel also observed that if instead of the

dyadic expansion we consider t-s.dk. expansion (t = 2,3,...) of x G [0,1] and

Nn(x,s,t) (s = 0,1,2,...,t - l,t = 2,3,...) is the number of s's among the

first n digits of the 4-adic expansion of x, then

Nn{x,s,t)lim

n—»oo

= 0 (s = 0,l,2,...,*-l;* = 2,3,...) D.2)n t

for almost all x. Hence Borel introduced the following:

Definition. A number x G [0, l] is normal if for any s = 0,1,2,..., t — 1; t =

2,3,... D.2) holds.

The above result easily implies

THEOREM 4.2 (Borel, 1909).

Almost all x G [0,1] are normal.

It is interesting to note that in spite of the fact that almost all x G [0,1] are

normal it is hard to find any concrete normal number.

Proof 1 of D.1). (Gap method). Clearly (cf. B.4))

En-'S,, = 0, En�S2 = n�.

Hence by Chebyshev inequality for any e > 0

Pdn^Snl >e}< n~xe-2

and by Borel - Cantelli lemma 1

n~2Sn3 —*¦ 0 a.s. (n —> oo).

Now we have to estimate the value of Sk for the fc's lying in the gap, i.e. between

n2 and (n + lJ. If n2 < A: < (n + IJ then

lAT^I = \n-*Sn*n*k-x + A:�E, - Sn,)| < \n~2Sn,\ + k~l{{n + lJ - n2).Since both members of the right hand side tend to 0, the proof is complete.

Proof 2 of D.1). (Method of high moments). A simple calculation gives

and again the Markov inequality and the Borel - Cantelli lemma imply the the-

theorem.

As we will see later on most of the proofs of the strong theorems are basedon a joint application of the above two methods.

30 CHAPTER 4

4.3 Between the Strong Law of Large Numbers

and the Law of Iterated Logarithm

The Theorem of Borel claims that the distance of the particle from the originafter n steps is \Sn\ = o(n) a.s. It is natural to ask whether a better rate can be

obtained. In fact we have

THEOREM OF HAUSDORFF A913). For any e > 0

lim n-1/2-eSn = 0 a.s.n—»oo

Proof. Let K be a positive integer. Then a simple calculation gives

ES2* = 0{nK).

Hence the Markov inequality implies

> nK+eK} < 0{n~eK).

If K is so big that eK > 1 then by the Borel - Cantelli lemma we obtain the

theorem. (The method of high moments was applied.)Similarly one can prove

THEOREM 4.3

lim sup .

n

<1 a.s.

n—oo n1/2logn

Proof. By B.4) we have

Eexptn-^S^e1/2 (n^oo).

Hence

= P{exp(n-1/25n) > exp((l + e) logn) = n1+e} < n��/

if n is big enough. Consequently

lim sup .

w<1 a.s.

n—oo n^'logn

and the statement of the Theorem follows from the symmetry of Sn.The best possible rate was obtained by Khinchine. His result is the so-called

Law of Iterated Logarithm (LIL).

STRONG LAW AND LIL 31

4.4 The LIL of Khinchine

LIL OF KHINCHINE A923).

Iimsup6n5n = Iimsup6n|5n| = limsup6nMnn—»oo n—»oo n—»oo

= \imsup bnM+ = limsup6nM~ = 1 a.s.

n—»oo n—»oo

where bn = Bnloglogn)~1/2.

Proof. The proof will be presented in two steps. The first one gives an upper

bound of lim supn_oo bnMn, the second one gives a lower bound of lim supn_oo bnSn.These two results combined imply the Theorem.

Step 1. We prove that for any e > 0

limsup6nMn < 1 + ?¦ a.s.

n—oo

By B.16) we obtain

P{Mn > (l + e)^1} <exp(-(l + ?:)loglogn) = (logn)-1� D.3)

if n is big enough. Let nk = [0*] @ > 1). Then by the Borel - Cantelli lemma

we get

Mnk < A + e)b~l a.s.

for all but finitely many k. Let n^ <n < njk+1. Then

Mn < Mnt+l < A + e)b~lk+l < A + 2s)b-lk < A + 2e)b~l a.s.nt+l

< A + e)bk+l <

provided that 0 is close enough to 1.

We obtain

limsup6nTn < 1n—»oo

where Tn is any of Sn, | Sn |, Afn, Af+, Af".

Observe that in this proof the gap method was used. However, to obtain

inequality D.3) it is not enough to evaluate the moments or the moment gener-

generating function of Mn (or that of Sn) but we have to use the stronger result of

Theorem 2.13.

Step 2. Let nk = [0*] @ > 1). Then for any e > 0

" Snk) >l-s}>

32 CHAPTER 4

if A: is big enough. Since the events {bnk+1 {Snk+1-Snt) > A -e)} are independent,we have by Borel - Cantelli lemma 2

">• a.8.

Consequently

Applying the result proved in Step 1 we obtain

if A; and 0 are big enough. Hence limsupn_oo bnSn > 1 — e a.s. for any e > 0,which implies the Theorem.

Note that the above Theorem clearly implies

limsupSn = oo, liminfSn = —oo a.s.,n—oo

n~*°°

which in turn implies the Recurrence Theorem of Section 3.1.

For later references we mention the following strong generalization of the LIL

of Khinchine.

LIL OF HARTMAN - WINTNER A941). Let YUY2,... be a sequence ofi.i.d.r.v.'s with

= 0, E^2 = 1.

Then

limsup&n(Yi +Y2 + -" + Yn) = 1 a.s.

n—»oo

Remark 1. Strassen A966) also investigated the case EYi2 = oo. In fact he

proved that if Yi, Y2,... is a sequence of i.i.d.r.v.'s with EYi = 0 and EYj2 = oo

then

Iimsup6n|yi + Y2 H h Yn\ = oo a.s.

n—»oo

Later Berkes A972) has shown that this result of Strassen is the strongest possibleone in the following sense: for any function f(n) with limn_oo f(n) = 0 there

exists a sequence Y\, Y2,... of i.i.d.r.v.'s for which EYi = 0, EYt2 = oo and

lim 6n/(n)|yi =0 a.s.


Chapter 5

Levy Classes

5.1 Definitions

The LIL of Khinchine tells us exactly (in certain sense) how far the particlecan be from the origin after n steps. A trivial reformulation of the LIL is the

following:(i) for any s > 0

Sn < A + ^b'1 a.s. for all but finitely many n

and

(")Sn > A - ?)Kl i-o- a.s.

Having in mind this form of the LIL, Levy asked how the class of those

functions (or monotone increasing functions) /(n) can be characterized for which

Sn < f(n) a.s.

for all but finitely many n. (i) tells us that A + e)b~l is such a function for

any e > 0 and (ii) claims that A — e)b~l is not such a function. The LIL does

not answer the question whether b'1 is such a function or not. However, one

can prove that b'1 is not such a function but Bn(log log n + 3/2 log log log n)I/2belongs to the mentioned class. In order to formulate the answer of Levy'squestion introduce the following definitions.

Let {Y(t),t > 0} be a stochastic process then

Definition 1. The function ax(t) belongs to the upper-upper class of {Y(t)} (at €

UUC(y(*))) if for almost all w € ft there exists aio = to{u) > 0 such that

Y(t) <ai(<) if t> t0.

33

34 CHAPTER 5

Definition 2. The function a2{t) belongs to the upper-lower class of

(a2 G ULC(F(i))) if for almost all w G 0 there exists a sequence of positivenumbers 0 < tv = ti(u) < t2 = t2{uj) < ... with tn —*• oo such that Y(ti) >

Definition 3. The function a3(t) belongs to the lower-upper class of

(a3 G LXJC(Y(t))) if for almost all u G Cl there exists a sequence of positivenumbers 0 < tx = ti[u) < t^ = ^(w) < ••• with tn —+ oo such that

Definition 4. The function aA{t) belongs to the lower-lower class of

(a4 G LLC(F(?))) if for almost all u G n there exists a t0 = to(uj) > 0 such that

Y{t) > aA{t) \it > t0.

Let Yx, F2,... be a sequence of random variables then the Levy classes UUC(Yn),ULC(rn), LUC(rn), LLC(Fn) of {Yn} can be defined in the same way as it was

done above for Y(t).We introduce two further definitions strongly connected with the above four

definitions of the Levy classes.

Definition 5. The process Y(t) is asymptotically deterministic (AD) if there

exists a function ax{t) G UUC(F(*)) and a function a4(t) G LLC(Y(t)) such that

limt-,00 |a4@ -ai(*)| = 0.

Consequently

lim |a4m - Y{t)\ = lim \ai{t) - Y(t)\ = 0 a.s.t—»oo t—*oo

Definition 6. The process Y(t) is quasi AD (QAD) if there exists a function

ax(<) G U\JC(Y(t)) andafunction aA{t) G LLC(F(*)) such that limsup^^ \aA{t)-ai{t)\ < oo.

The definition of AD resp. QAD sequences of r.v.'s can be obtained by a

trivial reformulation of Definitions 5 and 6.

Remark 1. Clearly the UUC(rn) resp. the XJXJC(Y(t)) is the complementerof the ULC(Fn) resp. the ULC{Y{t)) and similarly the LUC(rn) resp. the

LUC(y(*)) is the complementer of the LLC(yn) resp. the LLC(F(*)).

LEVY CLASSES 35

5.2 EFKP LIL

Now we formulate the celebrated Erdos A942), Feller A943, 1946), Kolmogorov- Petrowsky A930 - 35) theorem.

EFKP LIL The nondecreasing function a(n) € UUC(rn) if and only if

~a{n) ( a2{n)\^J < oo

n=l"

where Yn is any of n�/2S^n'1'2 \ Sn |,

This theorem completely characterises the UUC(Fn) if we take into consider-

consideration only nondecreasing functions and it implies

Consequence 1. For any e > 0

Sn < (nBloglogn + C + e) logloglogn)I/2 a.s. E.1)

for all but finitely many n. Further

Sn > (nBloglogn + 3logloglogn)I/2 i.o. a.s. E.2)

Here we present the proof of Consequence 1 only instead of the proof of EFKP

LIL (cf. Remark 1 at the end of Section 5.3).

Proof of Consequence 1. The proof will be presented in two steps.

Step 1. We prove that for any e > 0 and for all but finitely many n

Mn < (nB log log n + C + e) log log log n)I/2 a.s., E.3)

which clearly implies E.1). By B.16) we obtain

P{Mn > (nB log2 n + C + e) log3 n)I/2}

< 4A+4-1-! L_B_=3i+fLi-n *.. E.4)

V 2tt J2 log2 n lo8n (log2 n) a V ^ loSn (Iog2 n) 2

Let

f / * Mnjk = exp

Then by the Borel - Cantelli lemma we get

Mnk < {nkB log2 nk + C + e) log3 n^)I/2 a.s.

36 CHAPTER 5

for all but finitely many k. Let nk < n < nk+l. Then

Mn < Mnk+l < (nJk+1Blog2 nk+l + C + e) log3 n^)I'2< (nJkB log2 nk + C + 2e) log3 n*)I72 E.5)

which implies E.3).

Step 2. Introduce the following notations

- = [expy?(n) = Blog2n + 3log3nI/2,

<p*{n) = Blog2n + 6log3nI/2,An = {n

Then clearly

T(Ank) = Oik-1 (log k)-1),l(log k)-*'2),

and for any j < k — j + m we have

V(AnjAnk) = P {*>>,) > nJ1/J5Bj. > ^(n,),^172^ > <p(nk)}+ T{nJ1/2Snj > ^(n^nl^S^ > <p(nk)}

<V{tp*{ni)>nJ1/iSn.>ip{ni)

P

_[5ni- Sn. I n^XP {

'> J <p(nk) -

+ o (r^iogj)-*'*) = [o (rx(\ogjyy) - o

+ o(y-1(iogy)-5/2)where

and t =

-

n.

LEVY CLASSES

Observe that

and

Hence

Since

and

with

|3 log log j4

> <p{nk)

4 logj

'y/nk-Jnj y/*j 3loglogy"\y/nk

-

rij y/nk-

rij 4 log;

> i^3! if 1< x < 4fx-l 3

<x-\

x = — = exp

if x > 1

j \

= exp

for any j big enough, we obtain

m

m. log 1 + -

j V j

log; log(y + m

37

(-

rij-

3 \rij

1> -

~

3

m1/2

m. log 1 + —

j V j

iog(y + m) logy iog(y

1/21 ( m

4 \iogy

if 1 < m < (log 4) logj, and

1 m

1 ,> 1 —

exp—

m)

if m > (log4) logj.

38 CHAPTER 5

Similarly

1/2

Hence

? > O(m1/2) if 2 log log; < m < (log 4) log; E.6)

and4

^()if (l4) li < m < (i°gi) log log y.

In case m > log j(log log./) we obtain

and;+log;(loglog;)

? P(An>AnJ < O (j-'ilogj)-1 log log j) . E.8)

Having E.7) and E.8) a simple calculation gives

P(AnyAnJ=o((loglogiVJ)

arid

EP(il»J=O(loglogJV).

Hence the Borel - Cantelli lemma 2* of Section 4.1 and the Zero-One Law of

Section 3.2 imply the theorem. A simple consequence of EFKP LIL is

THEOREM 5.1 The nonincreasing function -c(n) € LLC(n-1/2S'n) if and

only if Ix{c) < oo.

The Recurrence Theorem of Section 3.1 characterizes the monotone elements

of LLC(|5n|). In fact

THEOREM 5.2 A monotone function d(n) € LLC(|5n|) if and only if d(n) < 0

for any n big enough.

Remark 1. For the role of Kolmogorov in the proof of EFKP LIL see BinghamA989).

LEVY CLASSES 39

5.3 The laws of Chung and Hirsch

The characterization of the lower classes of Mn and M+ is not trivial at all. We

present the following two results.

THEOREM OF CHUNG A948). The nonincreasing function a(n) e

LLC(n-^M,,) if and only if

h{a) = f; »-'(a(n))-'exp (-?«"») < <*>•

THEOREM OF HIRSCH A965). The nonincreasing function 0(n) €

LLC(n�/2M+) if and only if

Note that Theorem of Chung trivially implies

«-»«> \ n J V8

E.9) is called the "Other LIL".

Remark 1. The proof of EFKP LIL is essentially the same as that of Con-

Consequence 1. However, it requires a lemma saying that if a monotone function

/(•) E UUC(Sn) then f{n) > 6;1 /2 and if /(•) € ULC(S'n) then f(n) < 2b~x.

The proofs of Theorems of Chung and Hirsch are also very similar to the above

presented proof (cf. Consequence 1 of Section 5.2). However, instead of B.15)and B.16) one should apply B.17) and B.18).

5.4 When will Sn be very big?

We say that Sn is very big if Sn > b~l. EFKP LIL of Section 5.2 says that Sn is

very big i.o. a.s. Define

a{n) = max{k : 0 < k < n, Sk > &*1}, E.10)

i.e. <x(n) is the last point before n where Sk is very big. The EFKP LIL also

implies that a(n) = n i.o. a.s. Here we ask: how small can a(n) be? This

question was studied by Erdos - Revesz A989). The result is:

40 CHAPTER 5

THEOREM 5.3

(log log n)'/2 a(n)log =-G a.s.

n"-*00 (log log log n) log n n

where C is a positive constant with

ry-2 < fi < ol4

Equivalentlya(n) > nx~Sn a.s.

for all but finitely many n where

log log log n

n=(log log nI/*-

The exact value of C is unknown.

Clearly one could say that Sn is very big if

(i) Sn> (l-e)BnloglognI/2 @ < e < 1) or

(ii) Sn > Bn(log log n + | log log log n)) , e.t.c.

These definitions of "very big" are producing different a's instead of the one

defined by E.10). It is natural to ask: what can be said about these new a's?

It is also interesting to investigate the time needed to arrive from a very bigvalue to a very small one. Introduce the following notations: let

a.\ = min{A;: k > 3, Sk > b^1},01 = min{A;: k > a^, S^ < —b^1},oli = min{fc '• k > 0i, Sk > bfr },

02 — min{A;: k > a-i, Sk < — b^1},...

Define a sequence of integers {nk} by

_ c », r». *+! l iu o q ^n\ — o, n/c — itxjfc—i-x j \"'

—

"> **> • • •)•

Then by Theorem 5.3 between n^ and nk+i there exist integers j and / such that

Sj > bj1 and Si < -6,�.

Hence we have

LEVY CLASSES 41

THEOREM 5.4

0k < nk a.s.

for all but finitely many k.

Very likely a lower estimate of 0k is also close to nk but it is not proved.

Remark 1. The limsup of the relative frequency of those i's A A — eNfrl IS investigated in Section 8.1.

5.5 A theorem of Csaki

The Theorem of Chung (Section 5.3) implies that with probability 1 there are

only finitely many n for which

or in other words there are only finitely many n for which simultaneously

/ 2 \ i/2 / 2 \ V2

(-^M and8loglogn; ^y

for any 0 < e < 1. At the same time Theorem of Hirsch (Section 5.3) impliesthat with probability 1 there are infinitely many n for which

B\ !/2

( 8 log log n

In fact there are infinitely many n for which

AC < ?

Roughly speaking this means that if M+ is small (smaller than A—?){8i*\" n)then M~ is not very small (it is bigger than A - e)(81o^gwI/2 ) provided that

n is big enough. Csaki A978) investigated the question of how big M~ must be

if M+ is very small. His result is

THEOREM 5.5 Let a{n) > 0,6(n) > 0 be nonincreasing functions. Then

<a(n)n^ and M~ <b{n)n^ i.o.} = \\ ^ A(a(n),6(n)) = oo,n ~ v ' J

[ 0 otherwise

42 CHAPTER 5

where

and c(n) = a(n) + b[n).

The special case a(n) = b(n) of this theorem also gives Chung's theorem. For-

Formally this theorem does not contain Hirsch's theorem.

In order to illustrate what Csaki's theorem is all about we present here two

examples.

Example 1. Put

a{n) =C(loglogn)-1'2 @ < C < ir/y/s)

and

b{n) = ?>(loglogn)-1/2 (D > 0).

Then

J4(a(n), 6(n)) < oo if D < tt/v^ - C

and

/4(a(n), 6(n)) =oo if D > Ttjyfc - C.

Applying Csaki's theorem this fact implies that the events

(\1/2 / \ 1/2

—=—) and M-<D[-^—)loglogn/

n

Vloglogn/

occur infinitely often with probability 1 if D > n/v2 — C. However, it is not so

if D < ir/y/2 — C. That is to say if n is big enough and

/ \i/J

: <c[ —— (o < c < ifiVs),\loglogn/

then it follows that

'n —

/ \ 1/2

> D I for any 0 < D < tt/v^ - C.

\\og\ognj

Example 2. Put

a(n) = (logn)-a @ < a < 1)and

b{n) = J?;(loglogn)-1/2 {E > 0).

LEVY CLASSES 43

Then

J4(a(n),6(n)) < oo if 0<a<l and E < ttBA - a))~1/2and

J4(a(n),6(n)) =00 if 0< a < 1 and E > ttBA - a))�/2.Observe also that

^((lognJ-^FOogloglogn)-1/2) < oo if F<ir/V2and

J^OognJ^FOogloglogn)-1/2) = oo if F

Applying Csaki's theorem this fact implies that the events

{M+ <n1/2(logn)-a and M~ <

resp.

{M+ < n^logn)-1 and M" < Fn^logloglogn)�/2}occur infinitely often with probability one if E > 7rB(l — a))�/2 resp. F > n/y/2.However, it is not so if E < 7rB(l - a))�/2 resp. F < n/y/2.

As we have mentioned already, Csaki's theorem states that if one of the

r.v.'s M? and M~ is very small than the other one cannot be very small. It is

interesting to ask what happens if one of the r.v.'s M+ and M~ is very big. In

Section 8.1 we are going to prove Strassen's Theorem 1, which easily implies that

for any e > 0 the events

{Mn+>i^BnloglognI/2 and M" > ^BnloglognI/2}occur infinitely often with probability 1, but of the events

{Mn+>^BnloglognI/2 and M" > ^BnloglognI/2}only finitely many occur with probability 1. In general one can say

THEOREM 5.6 For any e > 0 and l/3<q<l the events

: > A - e)gb-1 and M~ > A - <0^Voccur infinitely often with probability 1, but of the events

x and M" > A+ e)ionly finitely many occur with probability 1.

44 CHAPTER 5

As a trivial consequence of Theorems 5.5 and 5.6 we obtain

THEOREM 5.7 Consider the range M*n = M++M~ of the random walk{Sn}.Then for any e > 0 we have

(l + e)BnloglognI/2 € UUC(M^),

A - e)BnloglognI/2 € ULC(M^),'2ir n

8 log log n)

v '

\ 8 log log n)

Theorems 5.5 and 5.6 describe the joint behaviour of M+ and M~. We also

ask what can be said about the joint behaviour of Sn and M~ (say). In order to

formulate the answer of this question we introduce the following notations.

Let 7{n),6{n) be sequences of positive numbers satisfying the following con-

conditions:

7(n) monotone,

6{n) | 0,

) T oo,

nxl26{n) T oo.

Further let f(n) = nl/2tjj{n) ? ULC(S'n) with t/>(n) | oo. Define the infinite

random set of integers

Then we have

THEOREM 5.8 (Csaki - Grill, 1988). For any f(n) = nx^{n) € ULC(Sn)the function

g{n) = nx'h{n) € UUC(M",n € f)

f{n)+2g{n)e\JXJC{Sn).

nx/2S{n) €LLC(Mn-,n€<r)

w-pf-^<-.

if and only

Further,

if and only

if

ifOO

En=l

n1/2(

*(»)n

LEVY CLASSES 45

Remark 1. nl/2-y(n) e UUC(M~,n e c) means that nl/2-y(n) > M~ a.s. for

all but finitely many such n for which n € <;. In other words the inequalitiesSn > f{n) and M~ > n1/'27(n) simulteneously hold with probability 1 only for

finitely many n.

Consequence 1. Let V(n) = min(M+,M"). Then /(n) € UUC(Vr) if and only

if3/(n) €UUC(S'n).

Remark 2. Theorem 5.6 in case q = 1/3 follows from the above Consequence1. For other g's A/3 < q < 1) Theorem 5.8 implies Theorem 5.6.

Example 3. Let f(n) = (B - ^nloglognI/2 @ < e < 2). Then we find the

inequalities

hold with probability 1 only for finitely many n. However,

/ e\ xl2 1 — e ( ( e\1/'2\Sn > A J 6� and M~ > I 1 — A 1 I b~l i.o. a.s.

The above two statements also follow from Strassen's theorem 1 (cf. Section 8.1).Further,

Sn>(l--) b-1 and M" <n1/2(logn)-"/2 i.o. a.s. E.11)

if and only if rj < e.

Chapter 6

Wiener process and Invariance

Principle

6.1 Two lemmas

Clearly the r.v. a~1(S(k + a) — S(k)) can be considered as the average speed of

the particle in the interval [k,k + a). Similarly the r.v.

a-vIx{n,a) = a~x max {S{k + a) - S{k))

is the largest average speed of the particle in @, n) over the intervals of size a.

We know (Theorem 2.9) that a~xl'l{S{k + a) - S{k)) is asymptotically (a -> oo)an JV@,1) r.v. Hence S(k + a) - S(k) behaves like a1/2 or by the LIL of Khinchine

S(k + a) - S(k)hmsup-^— L

tttt"= ! a-s-

a^oo Ba log log aI/2for any fixed k. We prove that even Ii(n, a) cannot be much bigger than a1/2. In

fact we have

LEMMA 6.1 Let a = an < na @ < a < 1). Then

limsup , )'

. < C a.s.

n—oo na/2(lognI/2if C > 4.

Proof. By Theorem 2.10 for any k

P

(c2\

logn) = n"c2/2

47

48 CHAPTER 6

as n —+ oo. Hence

f )

and the Borel - Cantelli lemma implies the statement.

Remark 1. Much stronger results than that of Lemma 6.1 can be found in

Section 7.3.

LEMMA 6.2 Let {XfJ; * = 1,2,...;/ = 1,2,...} be a double array of i.i.d.r.v. 's

with

EXfJ- = 0, EXj = 1, Eexp(*XfJ) < oo

for all t in some interval \ t |< tQ. Then for any K > 0 there exists a positiveconstant C = C(K) such that

for all but finitely many i.

Proof of Lemma 6.2 is essentially the same as that of Lemma 6.1 using Theorem

2.11 instead of Theorem 2.10.

6.2 Definition of the Wiener process

The random walk is not a very realistic model of the Brownian motion. In fact the

assumption, that the particle goes at least one unit in a direction before turningback, is hardly satisfied by the real Brownian motion. In a more realistic model

of the Brownian motion the particle makes instantaneous steps to the right or to

the left, that is a continuous time scale is used instead of a discrete one.

In order to build up such a model assume that in a first experiment we

can only observe the particle when it is located in integer points and further

experiments describe the path of the particle between integers. Let

{S(n) = SW(n), n = 0,1,2,...}

be the random walk which describes the location of the particle when it hits

integer points. Then we define a new Brownian motion S^^n) which is a "re-

"refinement" of S^(n), i.e. S^^n) is a random walk with the properties:

(i) the particle moves 1/2 to the right or to the left with probability 1/2,

WIENER PROCESS AND INVARIANCE PRINCIPLE 49

(ii) the time needed for one step is 1/4,

(iii) S^(n) hits the integers in the same order as S^(n) does.

Observe that for S^(n) and S^(n) the expectations of the waiting times to hit

a given integer are equal to each other.

In order to construct a random walk which satisfies the above three conditions

let

{Sk{n) = Sk>1 (n), n = 0,1,2,...} {k = 1,2,...)be a sequence of independent random walks. Sk(n) governs the moving of the

particle between S(k — 1) and S(k).Introduce the following notations:

To = fo.i = 0,

rk = rkA = inf{j : |Sfc(y)| =2} [k = 1,2,...),k

Tk = Tktl = Y,n,i {k = 0,1,2,...),

ak = sign^S^) - S{k - l))Sk{Tk - Tk^)),

S{1) (^) = S{k - 1) + l-akSk{n - Tk.x) if TM < n < Tk {k = 1,2,...).

Observe that

where

A) = iand t0 = 0, tj, t2,... is a sequence of integers with \ix — io\ — |t2 — *i||»n+i — t'n| = 1. In other words

i.e. the finite dimensional distributions of S^(n/4) are equal to the correspond-corresponding distributions of S(n)/2. This result can be formulated as follows: S^(n/4)is a Brownian motion with the property that the particle moves 1/2 to the rightor to the left with probability 1/2 and the time needed for one step is 1/4. Since

(* = 0,1,2,...)

50 CHAPTER 6

we say that S^ is a refinement of

We define similarly the refinement S^(n/16) of S'A)(n/4). Hence we define

a sequence {Sk,2{n),n = 0>l>2,...} of independent random walks, being also

independent from the previously defined random walks. Now let

if 7*_i,j < n < r4i2 (k = 1,2,...) where

at" = sign ((sC> g) - S<" (^)) SW(T« - 7i-,,)) ,

A:

T^.2 = ?>,i2 (A; = 0,1,2,...),i=o

7-0,2 = 0,

It can be easily seen that

and

Hence we say that S^(n/16) is a Brownian motion with the property that the

particle moves 1/4 to the right or to the left with probability 1/2 and the time

needed for one step is 1/16. Further, S^ is a refinement of S^\

Continuing this procedure we obtain a sequence of random walks

as follows: having {$(m> (n2�m), n = 0,1,2,...} defined, (S'(m+1)(n2-2m-2), n =

0,1,2,...} will be denned by:

_ Mm) (k ~ l\ ,L>)c (T \

if Tk-i,m+i <n< Tk,m+i (fc = 1,2,...) where

4"> = sign ((s<"> (A] _ S(~) fci)Vk

Tk,m+l = ^Ti.m+l (fc = 0,1,2, ...),1=0

T0,m+l = 0,

ri,m+i = inf{j : |5«,m+i(i)| =2} (/ = 1,2,...)

and {SiiTn+i(n),n = 0,1,2,...} (/ = l,2,...;m = 1,2,...) is a double array of

independent random walks.

It can be easily seen again that

and

= S<'V22-+2; \22-;-

Hence we say that S(m+1)(n2"�m~2) is a Brownian motion with the property that

the particle moves 2~(m+1) to the right or to the left with probability 1/2 and

the time needed for one step is 2"Bm+2). Further, S'(m+1) is a refinement of S^m\

A simple calculation givesEr,,w = 4,

and for any t0 > 0 there exists a C = C(t0) such that

Eexp(*r«im) <C

if \t |< t0.

Applying Lemmas 6.1 and 6.2 we find that for any K > 0 there exists a

C = C(K) > 0 such that

sup |S'(m+1)(A;2-2m) - S'(m)(A;2-2m)| < Cm2"m/2 a.s.

k2~'2m<K

for all but finitely many m. Hence as m —> oo the sequence

(A;2�m < t < (k + lJ�m) converges uniformly in any finite interval [0, T] to a

continuous function W(t) = W(t,u) for almost all u ? ft. This limit process is

called a Wiener process.

It is also easy to see that this limit process has the following three properties:

(i) W(t) - W(s) e N{Q,t- s) for all 0 < s < t < oo and W{0) = 0,

(ii) W{i) is an independent increment process that is ) )W(t4) - W(t3),..., W(t2i) -W{t2i.x) are independent r.v.'s for all 0 < tx <

ti <t3<U<...<t2i.1 <t2i (t = 2,3,...),

(iii) the sample path function W(t,u) is continuous in t with probability 1.

(i) and (ii) are simple consequences of the central limit theorem (Theorem 2.9).(iii) was proved above.

52 CHAPTER 6

6.3 Invariance Principle

Define the sequence of r.v.'s 0 < rt < r2 < ... as follows:

i-! =inf{t:t>O,\W{t)\ = l},r2 = mf{t : t > r^lWit) -WirJl = 1},

ri+l = \nf{t :t > r^lWit) -W(rt)\ = 1},...

Observe that

CO ^{ri),W{T2) - W{tx),W{tz) - W{t2),... is a sequence of i.i.d.r.v.'s with

distribution P{V^(r1) = 1} = TiW^) = -1} = 1/2, i.e. {W{Tn)} is a

random walk,

(ii) ri» r2~~

T\, Tz—

r2,... is a sequence of i.i.d.r.v.'s with distribution

Applying the reflection principle (formulated for Sn in Section 2.1, Proof 2 of

B.5)) for W(-) we obtain

'* °U~

) dt. F.1)

Evaluating the moments of rt and applying the strong law of large numbers we

obtain

Er! = 1, Er.2 = 2, lim n"Vn = 1 a.s.1

n—»oo

The above two observations are special cases of a theorem of Skorohod A961).Because of (i) we say that a random walk can be embedded to a Wiener process

(by the Skorohod embedding scheme).Applying the LIL of Hartmann - Wintner A941) (Section 4.4) and some

elementary properties of the Wiener process (formulated in Section 6.1) we obtain

\W(rn)-W(n)\hmSUp —

r-rr-T- -rjr < OO a.S.n-oo (n log log n) V4( log n) V2

This result can be formulated as follows:

THEOREM 6.1 On a rich enough probability space {fi, /, P} one can definea Wiener process {W(t),t > 0} and a random walk {Sn, n = 0,1,2,...} such that

\Sn-W(n)\hmsup j—— —j-t- —T- < oo a.s.

n-oo (n log log nI/4(log nI'2

This result is a special case of a theorem of Strassen A964).A much stronger result was obtained by Komlos - Major - Tusnady A975-76).

A special case of their theorem runs as follows:

INVARIANCE PRINCIPLE 1 On a rich enough probability space {ft, /,P}one can define a Wiener process {W(t),t > 0} and a random walk {Sn, n =

0,1,2,...} such that

\Sn-W{n)\ = O{logn) a.s.

Remark 1. A theorem of Bartfai A966) and Erdos - Renyi A970) impliesthat the Invariance Principle 1 gives the best possible rate. In fact if {Sn, n =

0,1,2,...} and {W(t),t > 0} are living on the same probability space {fi, /,P}then | Sn - W(n) |> O(logn) a.s.

As a trivial consequence of Invariance Principle 1 we obtain

THEOREM 6.2 Any of the EKFP LIL, the Theorems of Chung and Hirsch

and Theorems 5.3, 5.4 and 5.5 remain valid replacing the random walk Sn bya Wiener process W(t). As an example we mention: Let Y(t) be any of the

processes

rl'*\w[t)\,t~Xl2m(t\ — t~Xl2 <;iin IWMI6 TJX 16 1 — 6 otl^J I v /1 ?

Q<s<t

t~l/2rn+(t) = t~1/2 sup WE),Q<s<t

+-1/2rn~(t) — —t~xl2 Jnf W( <i\L TJX 16 1 ——

6 1111 ww loliv '

0<$<tv '

Then a nondecreasing function a(t) € V\JC(Y(t)) if and only if

r°° a(t) ( a2(t)\I exp I I at < oo. ("-^JJ\ t \ 2 )

Similarly a nonincreasing function [a{t))~x € \AjC{t~xl2m{t)) if and only if

\dt<oc. F.3)

Remark 2. In fact the EFKP LIL only implies that a(n) e UUC(F(n)) (n =

1,2,...) if a(-) is nondecreasing and F.2) is satisfied. In order to get our Theorem

6.2 completely we have to know something about the continuity of W(-), i.e.

54 CHAPTER 6

we have to see that the fluctuation supfc<tsupJk<J<Jk+1 \W(s) — W(k)\ cannot be

very big. For example, the complete result can be obtained by Theorem 7.13,especially Example 2 of Section 7.2, which says that the above fluctuation is

asymptotically Blog*I/2 a.s.

Theorem 6.2 claimed that any of the strong theorems formulated up to now

for Sn will be valid for W(-). The same is true for the limit distribution theorems

of Section 2.2. In fact we have

THEOREM 6.3

^ u} = 2A - $(u)) (* > 0,u > 0),

? (-l)*exP U^f^l] dx1

Jfc=-oo

For later references we give a more general form of the Invariance Principle1:

INVARIANCE PRINCIPLE 2 (Komlos - Major - Tusnady, 1975-76). Let

F(x) be a distribution function with

f°° xdF(x) = 0, f°° x2dF{x) = 1— oo J—oo

'

etxdF(x) < oo | t \< t0

with some t0 > 0. Then, on a rich enough probability space {17,/,P}, one can

define a Wiener process {W(t),t > 0} and a sequence of i.i.d.r.v.'s Y\, Y2,... with

< x} = F(x) such that

\Tn - W(n)\=O(log n) a.s.,

where Tn = Yx + Y2 + • • • + Yn.

Chapter 7

Increments

7.1 Long head-runs

In connection with a teaching experiment in mathematics, T. Varga posed a

problem. The experiment goes like this: his class of secondary school children is

divided into two sections.

In one of the sections each child is given a coin which he then throws two

hundred times, recording the resulting head and tail sequence on a piece of paper.

In the other section the children do not receive coins but are told instead that

they should try to write down a "random" head and tail sequence of lengthtwo hundred. Collecting these slips of paper, he then tries to subdivide them

into their original groups. Most of the time he succeeds quite well. His secret

is that he had observed that in a randomly produced sequence of length two

hundred, there are, say, head-runs of length seven. On the other hand, he had

also observed that most of those children who were to write down an imaginaryrandom sequence are usually afraid of putting down head-runs of longer than

four. Hence, in order to find the slips coming from the coin tossing group, he

simply selects the ones which contain head-runs longer than five.

This experiment led T. Varga to ask: What is the length of the longest run

of pure heads in n Bernoulli trials?

A trivial answer of this question is

THEOREM 7.1

lim :—— = 1 a.s.N->oo lg N

where Zn is the length of longest head-run till N.

55


56 CHAPTER 7

Proof.

Step 1. We prove that

Iiminf-%>1 a.s. G.1)Noo lgjV

~

Let e < 1 be any positive number and introduce the notations:

t=[(l-?)\gN],

Uk = St(k+1) — Stk (k — 0,1

Clearly UQ, U1,..., Ujf are i.i.d.r.v's with

1

p(^ = t) = —.

2*

Hence

i>(uQ<t,ul<t,...,uN-<t) = (i-^and a simple calculation gives

~N

-I < OO

for any ? > 0. Now the Borel - Cantelli lemma implies G.1).

Step 2. We prove that

limsup:—^r<l a.s. G.2)N—•• Ik N

Let ? be any positive number and introduce the following notations:

u = [(l +

fc = Sk+U-Sk {k = 0,l,...,N-u

= \J{Vk = u}Jk=O

and let T be any positive integer for which Te > 1. Then

P(Vt = u) = 2"u,

consequentlyoo

) < N2~u and

INCREMENTS 57

Hence the Borel - Cantelli lemma implies

ZuTlimsup—-F < 1 a.s. G.3)

jk^oo \gkT '

Let kT < n < (k + l)T and observe that by G.3)

Zn < Z{k+l}r <{! + ?) \g{k + 1)T < A + 2e) \gkT < A + 2e) lgn

with probability 1 for all but finitely many n. Hence we have G.2) as well as

Theorem 7.1.

A much stronger statement is the following:

THEOREM 7.2 (Erdos - Revesz, 1976). Let {an} be a sequence of positivenumbers and let

n=l

Then

an € UUC(Zn) if A{{an}) < oo, G.4)an G ULC(Zn) if A{{an}) =00 G.5)

and for any e > 0

An = [lgn - lglglgn + lglge - 2 - e] e LLC(Zn). G.7)

Example 1. If 6 > 0 and

< = lg n + A + <5) lg lg n then A({<}) < 00.

Hence G.4) and G.7) together say that

\n< Zn< a*n for all but finitely many n} = 1.

Note that if n = 223° = 21048576 ~ 10315621 and e = 6 = 0,1 then An = 1048569

and < = 1048598.

Remark 1. Clearly G.4) and G.5) are the best possible results while G.6) and

G.7) are nearly the best possible ones.

A complete characterization of the lower classes was obtained by Guibas - OdlyzkoA980) and Samarova A981). Their result is:

58 CHAPTER 7

THEOREM 7.3 Let

i^n = ^gn - \g\g\gn + \g\ge - 2.

Then

liminf[Zn - xjjn] = 0 a.s.n—»oo

It is also interesting to ask what the length is of the longest run containingat most one (or at most T, T = 1,2,...) (-l)'s. Let Zn(T) be the largest integerfor which

Il(n,Zn(T))>Zn(T)-2T

where

/i(n, a) = max {Sk+a - Sk).

A generalization of Theorem 7.2 is the following:

THEOREM 7.4 Let {an} be a sequence of positive numbers and let

n=l

Then

ane\J\JC{Zn{T)) if AT{{an}) < oo, G.8)an e ULC(Zn(T)) if ^T({an}) = oo G.9)

and for any e > 0

Kn{T) = [lgn + Tlglgn - lglglgn - lgT! + lglge - 1 + e]G.10)

Xn{T) = [lgn + Tig lgn- lglglgn- lgT'. + lglge- 2- e]eLLC(Zn(T)). G.11)

A trivial reformulation of the question of T. Varga is: how many flips are needed

in order to get a run of heads of size m? Formally speaking let Zm be the smallest

integer for which

As a trivial consequence of Theorem 7.2 we obtain

INCREMENTS 59

THEOREM 7.5

Am € UUC(Zm),Km e ULC(Zm),am e LUC(Zm) if A{{am}) = oo,

am e LLC(Zm) if A{{am}) < oo

where Km resp. Xm are the inverse functions of Km of G.6) resp. Xm of G.7) and

am is the inverse function of the positive increasing function dm.

Instead of considering the pure head-runs of size m one can consider any givenrun of size m and investigate the waiting time till that given run would occur.

This question was studied by Guibas - Odlyzko A980).Erdos asked about the waiting time Vm till all of the possible 2m patterns of

size m would occur at least once. An answer of this question was obtained byErdos and Chen A988). They proved

THEOREM 7.6 For any e > 0

lge

and

?)r ¦- e uuc(vm)mm

A much stronger version of this theorem was obtained by Mori A989). He proved


)-1 e UUC(Vt),)-1 e ULC(Vt),

{2kk - {1 - ?Jklglgk){lge)-1 e LVC{Vk),{2kk - {1 + sJk\g\gk){\ge)-1 e LLC{Vk).

We mention that the proof of Theorem 7.7 is based on the following limit distri-

distribution:

THEOREM 7.8 (Mori, 1989).

lim 2*/18sup {2~kVk -

{< y) - e~e = 0.

60 CHAPTER 7

In order to compare Theorem 7.7 and Theorems 7.2 and 7.3 it is worthwhile to

consider the inverse of Vk. Let

Un = max{k : Vk < n).

Then Theorem 7.7 implies

Corollary 1. (Mori, 1989). For any ? > 0 we have

[lgn-lglgn-elge^^j < Un < flgn - lglgn + (l + e) lge^-^l a.s.

[ lgn J L 18n J

for all but finitely many n. Consequently Un is QAD.

Observe that Un is "less random" than Zn. In fact for some n's the lower and

upper estimates of Un are equal to each other and for the other n's they differ

by 1. Clearly Un < Zn but comparing Theorems 7.2, 7.3 and Corollary 1 it turns

out that Un is not much smaller than Zn.In Theorem 7.2 we have seen that for all n, big enough, there exists a block of

size An (of G.7)) containing only heads but it is not true with Kn (of G.6)). Now

we ask what the number is of disjoint blocks of size An containing only heads.

Let un(k) be the number of disjoint blocks of size k (in the interval [0, n])containing only heads, that is to say un(k) = j if there exists a sequence 0 < ti <

ti + k < t2 < t-i + k < ... < tj < tj + k < n such that

Stt+k-St! = k (i = 1,2 j)

but

Sm+k ~Sm<k if U + k < m < ti+l (i = 1,2,... ,j - 1)

or tj + k < m < n — k.

The proof of the following theorem is very simple.

THEOREM 7.9 (Revesz, 1978). For any e > 0 there exist constants 0 < ax =

ai{?) 5: <*2 = Q=2(e) < oo such that

= hminf -—;—- < hmsup t—: = a2 a.s.n—oo lglgn n—oo lglgn

{for Xn see G.7)).

This theorem says that in the interval [0, n] there are O(lg lg n) blocks of size Ancontaining only heads. This fact is quite surprising knowing that it happens for

INCREMENTS 61

infinitely many n that there is not any block of size An + 2 > Kn containing onlyheads.

Deheuvels A985) worked out a method to find some estimates of ai(e) and

a~{s). In order to formulate his results let Zn = Z^ and let Z™ >Z^]>... be

the length of the second, third, ... longest run of l's observed in Xi, X2,..., Xn.Then

THEOREM 7.10 (Deheuvels, 1985). For any integer r > 3 and

k > 1 and for any e > 0

G.12)

(lg2n+ + lgr_1n + lgrn) G ULC(Z), G.13)

[lgn-lg3n + lglge-l]eLUC(ZW), G.14)

[lgn - lg3n + lglge - 2 - e] G LLC(zW). G.15)

Remark 2. In case k = 1 G.14) gives a stronger result than G.6) but G.14)and G.15) together is not as strong as Theorem 7.3.

THEOREM 7.11 (Deheuvels, 1985). Let v e @,+oo) be given, and let 0 <

c[ < 1 < c" < oo be solutions of the equation

c-l-logc = -. G.16)

Then for any e > 0 we have

^1l G.17)G.18)

[lgn - lg3 n + lg2 e - \gc'v + e\ G LUC(Zi"lo«»Bl), G.19)[lgn - lg3 n + lg2 e - lg< - 2 - e) G LLC(Z^°*^). G.20)

Remark 3. This result is a modified version of the original form of the theorem.

It is also due to Deheuvels (oral communication).Theorem 7.2 also implies that

liminf i/n(/n) = 0 a.s.

if ln > Xn but

.. ...

,,... .

n f = 0 if 6 > 0,limsupi/n([lgn+(H-«)lglgn])< f

.

Now we are interested in limsup,,^^ ^n([lg n + lg lg n]) an(l formulate our simple

62 CHAPTER 7

THEOREM 7.12 (Revesz, 1978).

limsupi/n([lgn +lglgn]) < 2 a.s. G.21)n—»oo

Finally we mention a few unsolved problems (Erdos - Revesz, 1987).

Problem 1. We ask about the properties of Zn - Z& = Z^ - Z?\ It is clear

that P{ZW = zW i.o. ) = 1. The limsup properties of Z^ - Z& look harder.

Problem 2. Let Kn be the largest integer for which

Characterize the limit properties of Kn. Observe that Theorem 7.9 suggests

0 < limsup -—: < oo.

n—oo log log U

Problem 3. Let Z^ be the length of the longest tail run, i.e. Z^ is the largestinteger for which

where

r(n,k)=Q<mm_k(Sj+k-S}).How can we characterize the limit properties of \Zn — Z^\l Note that by Theorem

7.2

limsup —— < 1 a.s.

n—oo log log Tl

and clearlyP{Zn = Z*n i.o.} = 1.

_|0 if Zn<Z*n,Un 11 if Zn>Z*n

Problem 4. Let

and

i.e. Un = 1 if the longest head run up to n is longer than the longest tail run. We

ask: does lim,,.^ ?n exist with probability 1? In the case when \im.n-tooXn = ?a.s. then ?. is called the logarithmic density of the sequence {Un}.

INCREMENTS 63

Problem 5. (Karlin - Ost, 1988). Consider two independent coin tossingsequences Xu X2,..., Xn and X[, X2,..., X'n. Let Yn be the longest common

"word" in these sequences, i.e. Yn is the largest integer for which there exist a

1 < kn< kn + Yn<n and a 1 < k'n < k'n + Yn < n such that

Xkn+j = Xk'r.+i if i = 1> 2, • • •, Vn-

Karlin and Ost A988) evaluated the limit distribution of Yn. Its strong behaviour

is unknown. Petrov A965) and Nemetz and Kusolitsch A982) investigated the

length of the longest common word located in the same place, i.e. they defined

Yn assuming that kn = k'n. In this case they proved a strong law for Yn.

7.2 The increments of a Wiener process

This paragraph is devoted to studying the limit properties of the processes

Mt,at)(i = 1,2,3,4,5) where at is a regular enough function (cf. Notations to the

increments).Note that the r.v. a~1(W(s + a) — W(s)) can be considered as the average

speed of the particle in the interval (s, s + a). Similarly the r.v.

a-1J1{t,a)=aT1 sup (W{s + a) - W{s))0<3<t-a

is the largest average speed of the particle in @, t) over the intervals of size a.

The processes J,(t,a)(i = 2,3,4,5,? > a) have similar meanings.Note also that

Ji{t,at) < min{J2{t,at),J3{t,at)}max{J2{t,at),J3(t,at)} < J4{t,at).

To start with we present our

THEOREM 7.13 (Csorgo - Revesz, 1979/A). Let at(t > 0) be a nondecreasingfunction of t for which

(i) 0 < at < t,

(ii) t/at is nondecreasing.

Then for any i = 1,2,3,4 we have

at) — W(t)\()t—»oo t—»oo

at) - W{t)) = 1 a.s.

t-*oo

64 CHAPTER 7

where

= [2at (log— + log log t

at

If we have also

(iii)lim (log — ) (log log t)

1=

t-K» \ at/

= oo

then

lim itJi(t,at) = 1 a.s.t—»oo

In order to see the meaning of this theorem we present a few examples.

Example 1. For any c > 0 we have

lf a.s. A = 1,2,3,4). G.22)t

This statement is also a consequence of the Erdos - Renyi A970) law of largenumbers.

Example 2.

lim ,:?_'*{l9 = 1 a.s. (x = 1,2,3,4). G.23)

Example 3. For any 0 < c < 1

In case c = 1 we obtain the LIL for Wiener process (cf. Theorem 6.2). Note that

G.24) is also a consequence of Strassen's theorem of 8.1.

Having Theorem 7.13 it looks an interesting question to describe the Levy-classes of the processes J,-(t,at)(t = 1,2,3,4) in case of different at's. Unfortu-

Unfortunately we do not have a complete description of the required Levy-classes. We

can only present the following results:

THEOREM 7.14 (Ortega- Wschebor, 1984). Let f(t) be a continuous nonde-

creasing function and assume that at satisfies conditions (i) and (ii) of Theorem

7.13. Then

f{t) e UUC (ar1/2J,.(*,at)) (r= 1,2,3,4)

INCREMENTS 65

2

Li?i«p _O!2 *<«». G.25)<*t \ 2 J

Further, if

I ^-^-exp I — I dt = oo G.26)

f{t) e ULC (ar1/2Ji(t,at)) (i = 1,2,3,4).

Remark 1. In case at = t condition G.26) is equivalent with the correspondingcondition of the EFKP LIL of Section 5.2. However, condition G.25) does not

produce the correct UUC in case at = t. Hence it is natural to conjecture that,in general, the UUC can be characterized by the convergence of the integral of

G.26). It turns out that this conjecture is not exactly true. In fact Grill A989)obtained the exact description of the upper classes under some weak regularityconditions on at. He proved

THEOREM 7.15 Assume that

at = Coexp( T —dy) < St

where 0 < 6 < 1, g(y) is a slowly varying function as y —> oo,Co,C1 are posi-positive constants.

Let f(t) > 0(t > 0) be a nondecreasing function. Then

f{t) e UUC (ar1/2(J,(*,at)) (*' = 1,2,3,4)

if and only if

< oo.

In order to illustrate the meaning of this theorem we present a few examples.

Example 4. Let at = {\ogt)a (a > 0). Then g(t) = a/log* and

( "-1 V"fp<e{t) = 2 log* + C - 2a) log21 + 2 ? logj. t + B + e) logp t

e UUC (ot�/2J,) if and only if ? > 0 (i = 1,2,3,4; p = 3,4,5,...).

66 CHAPTER 7

Example 5. Let at = exp((logOa) @ < a < 1). Then g(t) = a(logO"� and

/p,,@ = 2 log t - 2(log t)a + C + 2a) log21 + 2 ? log, t + B + e) logp t

e UUC (at�/2J,) if and only if ? > 0 (i = 1,2,3,4; p = 3,4,5,...).

Example 6. Let at = ta@ < a < 1). Then </(«) = a and

/p,«@ = I 2A - a) log t + 5 log21 + 2 ^ log, * + B + e) logp t

e UUC (at~1/2 Ji) if and only if e > 0 (i = 1,2,3,4; p = 3,4,5,...).

Example 7. Let at = at @ < a < 1). Then </(«) = 1 and

/ P-i

/p,«@ = 2 log21 + 5 log31 + 2 ? log, t + B + e) logpV i=*

e UUC (aTl/2J,-) if and only if e > 0 (i = l,2,3,4;p = 4,5,6,...).

Theorem 7.15 does not cover the case at/t —> 1. As far as this case is concerned

we present

THEOREM 7.16 (Grill, 1989). Let at = t(l - 0(t)) where 0{t) is decreasingto 0 and slowly varying as t —> oo and f(t) > 0 be a nondecreasing continuous

function. Then

f{t) e UUC (aJ^J^at)) {i = 1,2,3,4)

if and only if

< oo.

The characterization of the lower classes is even harder. At first we present a

theorem giving a nearly exact characterization of the lower classes when at is not

very big.

THEOREM 7.17 (Grill, 1989). Assume that

T

/\ct y

with(,. log*7(* . .

hmsup-——5-L < -«t-oo log log t

or

log log t/at_^

log log t

then

K1 = \ogn, K2 = \og-,4

log- < K3 < log7T,4

INCREMENTS 67

Then for any i — 1,2,3,4 we have

LUL. iCtj Jt[t, (It) j 1/ A < A,-

where

logTr < Ki < Iog47r,

log - < K-i < log it,47T

log- < ^3 < log47T,4

log — < ^4 < log 7T.

lb

// in addition either at is of the form

at = Coexp

^Kilogj.16 4

Remark 2. A very similar result was obtained previously by Revesz A982).However, some of the constants given there are not correct.

Example 8. Let at = «e-rlo«logt@ < r < oo). Then

A(t) = (exp(rloglogO)(loglog«)� T oo.

Hence.. .

c Ji{t,at)hminf t :—; tttt

= 1 a.s.t-oo Batrloglog«I/2

This result was proved by Book and Shore A978).If at is so big that the condition A(t) j oo does not hold the situation is even

more complicated. We have two special results (Theorems 7.18 and 7.19) only.

68 CHAPTER 7

THEOREM 7.18 (Csaki - Revesz, 1979, Grill, 1989). // A{t) = C > 0, i.e.

at = Ct(loglogt)� then with probability 1 we have

...... . / +oo if C < T,hminfJ1(*,at) =

j_oo lf C>T

where T is an absolute, positive constant, its exact value unknown.

If A(t) ->() then

lim inf —p , , =-/?( — ) a.s.«-« ^24 log log * V t /

and

Remark 3. Note that if at = at and I/a is integer then fi(at/t) = a. We

return to the discussion of this theorem in Section 8.1 in the special case when

at = at @ < a < l). The first part of Theorem 7.18 suggests the followingquestion. Does there exist a function at for which liminfj-^oo Ji(t,at) = 0 a.s.?

THEOREM 7.19 (Csaki - Revesz, 1979). // A{t) — 0 then

1 < liminf <5(«) J4{t,at) < 2>/2 a.s.t—*OO

where

(*2 r1/26(t) = [jJ

Remark 4. In case at = t, Theorem of Chung (Section 5.3) implies that

liminf 6(t)J4(t,at) = 1 a.s.

However, this relation does not follow from Theorem 7.19.

Remark 5. Ortega and Wschebor A984) also investigated the upper classes of

the "small increments" ofW(-). These are defined as follows:

Mt,at)= sup {W{s + as)-W{s)),0<3<t-a,

INCREMENTS 69

J7{t,at)= sup \W{s + a3)-W{s)\,0<s<t-at

JB(t,at) = sup sup (W(s + u)-0<s<t-at 0<u<a,

J9(t,at) = sup sup \W(s + u) - W(s)\0<s<t-a, 0<u<a.

where a, is a function satisfying conditions (i) and (ii) of Theorem 7.13.

Remark 6. Hanson and Russo A983/B) studied a strongly generalized version

of the questions of the present paragraph. In fact they described the limit pointsof the sequence

W{pk)-W{ak)B@k - ak)(\og@k/@k - ak)) +

for a large class of the sequences 0 < ak < /3k < oo.

Finally we present a result on the behaviour of J${-, •)¦

THEOREM 7.20 (Csorgo - Revesz, 1979/B). Assume that at satisfies condi-

conditions (i), (ii) of Theorem 7.13. Then

where

and

liminf Kt Js(t,at) = 1 a.s.t—»oo

Js{t,at)= inf sup \W(s + u)-W{s)Q<s<t-at o<u<at

'"I"// (iii) of Theorem 7.13 is also satisfied then

lim KtJs{t,at) = 1 a.s.t—>OO

The following examples illustrate what this theorem is all about.

Example 9. Let at = ^\ogt hence Kt -»• l{t -> oo). Then Theorem 7.20 tellsus that for all t big enough, for any e > 0 and for almost all a; € ft there existsa 0 < s = s{t,?,u) < t- at such that

sup \W(s + u) -W(s)\ < 1 + e

70 CHAPTER 7

but, for all s G [0,t — at\ with probability 1

sup \W{s + u)-W(s)\>l-?.

At the same time Theorem 7.13 stated the existence of an s G [0, t — at\ for which,with probability 1,

s+^-Xogt} -W{s)

but for all s e [0,t - at\

sup \W{s + u) -W{s) | <(- + e) log t.

Example 10. Let at = t. Then Theorem 7.20 implies

\1/2su ^t—oo y nH J 0<3<t

Hence we have the Other LIL (cf. E.9)).

Example 11. Let at = (log*I/2 hence Kt « ^(log01/4- Then Theorem 7.20

claims that for all t big enough, for any e > 0 and for almost all u G ft there

exists an s = s(t, e,uj) G [0, t — at] such that

supo<u<(iogtI/2

That is to say the interval [0,t — at] has a subinterval of length (logfI/2 where

the sample function of the Wiener process is nearly constant; more precisely, the

fluctuation from a constant is as small as A + ?Or8~1/2(logt)~1/4.This result is sharp in the sense that for all t big enough and all s ? [0, t — at]

we have with probability 1

sup \W{s + u) - WE)| > A - ?)^=(log0-1/4.0<u<(logtI/2 V°

Clearly, replacing the condition at = (log?I//2 in Example 11 by at = o(logt),we also find that there exists a subinterval of [0, t — at] of size at where the samplefunction is nearly constant. Csaki and Foldes A984/A) were interested in the

analogue problem when the term "nearly constant" is replaced by "nearly zero".

They proved

INCREMENTS 71

THEOREM 7.21 Assume that at satisfies conditions (i), (ii) of Theorem 7.13.

Then

liminf/it inf sup |W(s + u)| = 1 a.s.t-oo C0<5<t-a« o<u<a,

where

ir2at

If (iii) of Theorem 7.13 is also satisfied then

lim ht inf sup |W(s + u)| = l a.s.?-00 0<»<t-a«0<u|ac

' V "

Example 12. Letting at = t we obtain the Other LIL (cf. E.9)).

Example 13. If at = o(logt) then ht —> oo and

lim inf sup \W(s +u)| = 0t-ooO<J<t-a«0<u<a«

while in case at = 4c*7r~2 \ogt we have /it —» c� as ? —> oo and

lim inf sup \W(s + u)\ = c.

t-ooO<»<ta<|' V ;l

(Compare Example 13 in case c = 1 and c = \/2 with the first part of Example9.)

Theorems 7.13 - 7.19 gave a more or less complete description of the strongbehaviour of J,-(t, at)(i = 1,2,3,4). To complete this Section we give the followingweak law:

THEOREM 7.22 (Deheuvels - Revesz, 1987). Let t/at = dt. Assume that

lim dt = oo. G.27)

Then for any i = 1,2,3,4 in probability

G 2g)t—oo log log dt

We also mention that the proof of Theorem 7.22 is based on the following:

72 CHAPTER 7

THEOREM 7.23 (Deheuvels - Revesz, 1987). Assume G.27). Then for any

—oo < y < oo we have

—» exp(—e~v) (? —* oo)

t/ x = 2,3,4 and

-> exp(-e"») (t -> oo).

Note that in the above two theorems we have no regularity conditions on at

except G.27).In order to understand the meaning of G.28) consider the case t = 2 and

assume

-i^- = r @<r<oo). G.29)log log t

K ' K '

In this case Theorem 7.13 implies that J2{t, at) can be as big as

In the same case Theorem 7.17 implies that Ji{t,at) can be as small as

BatI/2(rloglog01/2.

G.28) describes the "typical" behaviour of J2(t,at) under the condition of G.29).Namely it behaves like

It is worthwhile to mention an equivalent but simpler form of Theorem 7.23.

THEOREM 7.24

limp] sup(WE + l)-WE)) <f{y,t)\ =exp{-e~v)t-*°° [0<3<t J

and

where

lim P I sup \W(s + l)-W (s) \<f(y,t)\= exp(-2e~v)

INCREMENTS 73

Let us give a summary of the results of this section. To study the properties of the

processes Ji(t,at) (i = 1,2,3,4,5) we have to assume different conditions on at.

For the sake of simplicity from now on we always assume that at is nondecreasingand satisfies conditions (i) and (ii) of Theorem 7.13.

Then the limit distributions of ./,(•,•) for i = 1,2,3,4 are given in Theorem

7.23. Observe that the limit distributions in case i = 1 and in case i = 2,3,4 are

different. The limit distribution of J5(-, •) is unknown. The exact distribution is

not known in any case.

A description of the upper classes of J,(-, •) (t = 1,2,3,4) is given in Theorem

7.14 but there is a big gap in this theorem between the description of UUC(J,) and

ULC(J,), i.e. there is a big class of very regular functions for which Theorem

7.14 does not tell us whether they belong to the UUC(J.) or to the ULC(J,).This gap is filled in by Theorem 7.15 if at satisfies a weak regularity condition.

However, this regularity condition excludes the case at/t —> 1. This case is

studied in Theorem 7.16. The above-mentioned results do not tell us anythingabout Js(-,-). In case if at is not very big (condition (iii) of Theorem 7.13 is

satisfied) a very weak result is given in Theorem 7.20.

The lower classes of J,(-, •) (i = 1,2,3,4) are "almost" completely described if

at <C tj log log t by Theorem 7.17. If at does not satisfy this condition Theorems

7.18, 7.19 resp. 7.20 tell something about the lower classes of Ji(-, •), J^-, •) resp.

Js(-, •) but we do not have a complete characterization and we do not have any

results (except trivial ones) about the lower classes of ^O,') and Js(-,-).

7.3 The increments of

By the Invariance Principle 1 (cf. Section 6.3) we obtain that any theorem of

Section 7.2 will remain true, replacing the Wiener process by a random walk (i.e.replacing J, by J,-(t = 1,2,3,4,5)) provided that '¦y� is big enough or equivalentlyan is big enough. In fact Theorems 7.13, 7.18 - 7.21 resp. 7.14 - 7.17 remain

true if an » logn resp. an » (lognK while Theorems 7.23 and 7.24 remain

true as they are. Hence we only study the increments of Sn in the case when

Hindoo an(logn)~3~e = 0 for any e > 0.

A trivial consequence of Theorem 7.1 resp. Theorem 7.13 (cf.also Example 1

of Section 7.2) is

THEOREM 7.25

lim/l("'lgn)=l a... G.30)n—oo lgn

74 CHAPTER 7

G.31)lgn n^°° logn

Remark 1. Comparing G.30) and G.31) we can see that the behaviours of 7iand Jx are different indeed if an = c logn@ < c < oo). As a consequence we also

obtain that the rate O(logn) of Invariance Principle 1 (cf. Section 6.3) is the

best possible one. This observation is due to Bartfai A966) and Erdos - Renyi

A970).Theorems 7.2, 7.3, 7.4 imply much stronger results on the behaviour of /,(-, •)

than G.30) of Theorem 7.25. In fact we obtain

THEOREM 7.26 Assuming different growing conditions on {an} we get

(i) // for some e > 0

on< [lgn-lglglgn + lglge-2-e] = An.

Then

Ii(n,an) = an a.s. (i = 1,2,3,4)

for all but finitely many n, i.e. 7,(n,an) is AD.

(ii) Let

An <an < [lgn + lglgn-lglglgn + lglge-2-<¦:] = An(l).

Then Ii(n,an)(i = 1,2,3,4) is QAD and 7,(n,an) = an or an— 2 a.s. for

all but finitely many n.

(iii) Let

An(l) <an< [lgn + lglgn + (l + e)lglglgn] = dn(l).Then 7,(n,an)(i = 1,2,3,4) is QAD and 7j(n,an) = an or an

— 2 or an— 4

a.s. for all but finitely many n.

(iv) In general, if

dn(T) = [lgn + Tig lgn + (l + e) lglglgn] < an < An(T + 1)= [lgn+(T+l)lglgn-lglglgn-lg((T + l)!) + lglge-2-e]

then 7j(n, an) is QAD and 7,(n, an) = an— 2T — 2 or an

— 2T, and if

An(T+l) <an<dn{T + l)

then 7,(n,an) is QAD and 7;(n,an) = an-2T-4 or an-2T-2 or an-2T.

INCREMENTS 75

The above theorem essentially tells us that Ii(n,an) is QAD with not more than

three possible values if an < lgn + Tiglgn for some T > 0. The next theorem

applies for somewhat larger an.

THEOREM 7.27 (Deheuvels - Erdos - Grill - Revesz, 1987). Let an = O(lgn)and 0 < T = Tan < an/2 be nondecreasing sequences of integers. Then

an-2TeLLC{Ii{n,an)) if ? exp(-2npBn)) < oo,n=l

an- 2T e LUC(/,(n, an)) if ? exp(-2npBn)) = oo,

n=l

an-2TeULC{Ii{n,an)) if f) 2>Bn) = oo,n=l

an -2TeUUC{It;(n,on)) t/ f) 2>Bn) < oo

n=l

tu/icrc

Here we present a few consequences.

Consequence 1. Let

an = lgn + /(n)be a nondecreasing sequence of positive integers with f(n) = o(lg n).

(i) Assume that

lim / ^n~= 0 for any e > 0.

n—oo (lgn)eThen 7,(n,an) is QAD and there exist an a.i(n) G UUC(/,(n,an)) and an

a4(n) G LLC(/,(n,an)) such that ai(n) — a^(n) < 3.

(ii) Assume that

/(n) = O((lgn)e) @<6<l).Then 7,(n,an) is QAD and there exist an a\{n) G UUC(/,(n,an)) and an

cn(n) G LLC(/,(n,an)) such that a^n) - a4(n) < ^ + 1.

(iii) Assume that

lim -—^— = oo for any e > 0.n~>0° (lgn)

Then /,(n,an) is not QAD.

76 CHAPTER 7

Consequence 2. Let an = [C lgn] with C > 1. Then

eJ/9lglgn€UUC(/i(n,an)),C(l-2/?)lgn+(l-eJplglgn€ ULC(/,(n,an)),

C(l - 2/3) lg n - 2p lg lg n - Ap lg lg lg n+

+ l + se LUC(/,(n,an)),

C(l-2/?)lgn-2plglgn-4plglglgn+4p lg(l - 2/3) + 4p lg lg c + 2p lg 7T + Gp + 1 - e € LLC^n, an))

where /? is the solution of the equation

I^)C = 2,

and e is an arbitrary positive number.

Remark 2. Consequence 2 above is a stronger version of an earlier result of

Deheuvels - Devroye - Lynch A986).In the case an » lg n we present the following:

THEOREM 7.28 (Deheuvels - Steinebach, 1987). Let a^ be a sequence ofpositive integers with an = [an] where an/logn is increasing and an(logn)~p is

decreasing for some p > 0. Then for any e > 0 we have

anan- tflogan + C/2 +*-)*;1 log log n G UUC(/,(n,an)),

anan- t~l logan + C/2 - e)t~l log log n € ULC(/,(n,an)),

anan-^1logan + (l/2 + e)^1loglogn € LUC(/,(n,an)),anan - t-1 \ogan + A/2 - e)t~l log log n € LLC(/,(n,an))

where an is the unique positive solution of the equation

exp(-(logn)/an) = A + an)A+

and1

,1 + an

tn = -log- .

2 1 -

an

Note that an » Ba;1 log nI/2.


INCREMENTS 77

In order to study the properties of 1$ resp.

lUn,an) = min max \Sj+i\,SK ' n)0<j<n-an 0<i<an

' } '

first we mention that by the Invariance Principle the properties of J$ resp.

j;(t,at) = inf sup0<«<«-a, o<u<at

will be inherited if an > (lognK+e(e: > 0). In fact Theorems 7.20 and 7.21 will

remain true if J5 resp. J5* are replaced by 75 resp./j and an > (lognK+e(e > 0).Hence we have to study the properties of 75 resp. /? only when an(log n)~3~e —>¦

0 (n —>¦ oo) for any e > 0. It turns out that Theorem 7.21 remains true if

an/ logn —>¦ oo (n —>¦ c»). In fact we have

THEOREM 7.29 (Csaki - Foldes, 1984/B). Assume that an satisfies condi-

conditions (i) and (ii) of Theorem 7.13 and

lim an(logn)� = oo.n—»oo

Then

lim inf hnlc(n,an) = 1 a.s.n—»oo

where hn is defined in Theorem 7.21. J/ condition (iii) o/ Theorem 7.13 is a/so

satisfied, then

lim /in/5*(n,an) = 1 a.s.n—»oo

If an = [c log n] then we have

THEOREM 7.30 (Csaki - Foldes, 1984/B). Let an = [c log n](c > 0) and definea* = a*(c) > 1 as the solution of the equation

if a*(c) is not an integer then

a.s.

for,all but finitely many n, i.e. II is AD,if a*(c) is an integer then

*{c)-l<r5{n,an)<a*{c) a.s.

for all but finitely many n, i.e. II is QAD. Moreover

75*(n,an) = a{c) - 1 i.o. a.s.

and

75*(n,an) = a*{c) i.o. a.s.

78 CHAPTER 7

The properties of 75 are unknown when log n <C an < (log nK. However, we have

THEOREM 7.31 (Csaki- F51des, 1984/B). Letan = [clogn\(c > 0) and definea = a(c) > 1 as the solution of the equation

¦K

cos -— =

if a(c) is not an integer then

h{n,an) = \a(c)} a.s.

for all but finitely many n, i.e. 1$ is AD,if a(c) is an integer then

a(c) — 1 < 75(n,an) < a(c) a.s.

for all but finitely many n, i.e. 1$ is QAD. Moreover

Is = a(c) — 1 i.o. a.s.

and

Is = a(c) i.o. a.s.

Chapter 8

Strassen type theorems

8.1 The theorem of Strassen

The Law of Iterated Logarithm of Khinchine (Section 4.4) implies that for any

e > 0 and for almost all uj ? Q there exists a random sequence of integers0 < i%i = rii{e,u>) <n-i — n2(e,u>) < ... such that

S{nk) > A -e)Bn4loglogn*I/2 = A - ej^n*))�. (8.1)

We ask what can be said about the sequence {Sj-,j = 1,2,..., nk} (provided that

(8.1) holds). In order to illuminate the meaning of this question we prove

THEOREM 8.1 Assume that nk = nk(e,u) satisfies (8.1). Then

S([nk/2}) > A - e)±S{n>) > ^^(ftK))� a.*. (8.2)


Proof. Let 0 < a < 1 — 2e and assume that

a^K))� < S{\nk/2\) < [a + e)[b{nk))-\ (8.3)

Then by (8.1)S(nk) - S([nk/2}) > (l - a - 2e)(b(nk))-\ (8.4)

By Theorem 2.10 the probability that the inequalities (8.3) and (8.4) simultane-

simultaneously hold is equal to O((lognik)-2(a2+A-a-2eJ)).Observe that if a / 1/2 and e is small enough then 2(a2 + A - a — 2eJ) > 1.

Hence by the method used in the proof of Khinchine's theorem (Step 1) we obtain

79

80 CHAPTER 8

that the inequalities (8.3) and (8.4) will be satisfied only for finitely many k with

probability 1. This fact easily implies Theorem 8.1.

Similarly one can prove that for any 0 < x < 1

S{[xnk})>{l-e)xS{nk) a.s. (8.5)


(8.5) suggests that if nk satisfies (8.1) and k is big enough then the process

{^([znjfc^O < x < 1} will be close to the process {xS(nk);0 < x < 1}. It is

really so and it is a trivial consequence of

STRASSEN'S THEOREM 1 A964). The sequence

sn{x) = bn{S[nx] +(x- I^H X[nx]+1) @ < x < 1; n = 1,2,...)

is relatively compact in C@, l) with probability 1 and the set of its limit points is

S (see notations to Strassen type theorems).The meaning of this statement is that there exists an event Qo C Q of prob-

probability zero with the following two properties:

Property 1. For any u> / Qo and any sequence of integers 0 < n^ < n2 < ...

there exist a random subsequence nk. = rik^u)) and a function / G S such that

snk,(x,uj) —> f(x) uniformly in x € [0,1].

Property 2. For any / G S and u ? Qo there exists a sequence of integersft* = rik(u>,f) such that

snk(x,u) —> f(x) uniformly in x G [0,1].

The Invariance Principle 1 of Section 6.3 implies that the above theorem is equiv-equivalent to

STRASSEN'S THEOREM 2 A964). The sequence {wn{x);0 < x < 1} is

relatively compact in C@,1) with probability 1 and the set of its limit points is Swhere

wn{x)=bnW{nx) (n = l,2,...).

Remark 1. Since |/A)| < 1 for any function / G S and f(x) = x G S, Strassen's

theorem 1 implies Khinchine's LIL.

STRASSEN TYPE THEOREMS 81

Consequence 1. For any e > 0 and for almost all u> ? Q there exists a To =

TQ(?,u>) such that if

W{T) > A - e){b{T)Yl for some T > To

then

sup <

Consequence 1 tells us that if W{t) "wants" to be as big in point T as it can

be at all then it has to increase in @, T) nearly linearly (that is to say it has to

minimize the used energy).The proof of Strassen's theorem 2 will be based on the following three lemmas.

LEMMA 8.1 Let d be a positive integer and a\, a^-, • • ¦ ,a-d be a sequence of real

numbers for which

1=1

Further, let

W*{n) = aiW{n) + a2{W{2n) - W{n)) + ¦¦¦ + ad{W{dn) - W{{d - l)n)).

Then

limsup6nW*(n) = 1 a.s. (8.6)n—»oo

and

liminf6nW*(n) = -1 a.s. (8.7)

Proof of this lemma is essentially the same as that of the Khinchine's LIL. The

details will be omitted.

The next lemma gives a characterization of S.

LEMMA 8.2 (Riesz - Sz.-Nagy, 1955, p.75). Let f be a real valued function on

[0,1]. The following two conditions are equivalent:

(i) / is absolutely continuous and Jo{f'Jdx < 1,

and f is continuous on [0, l].

82 CHAPTER 8

In order to formulate our next lemma we introduce some notations. For any

real valued function / € C@,1) and positive integer d, let f(d) be the linear

interpolation of / over the points i/d, that is

Sd = {/(d) : / e S}

where Sd C S by Lemma 8.2.

LEMMA 8.3 The sequence {w^(x);0 < x < 1} is relatively compact in Cdwith probability 1 and the set of its limit points is Sd-

Proof. By Khinchine's LIL and continuity of Wiener process our statement

holds when d = 1. We prove it for d = 2. For larger d the proof is similar

and immediate. Let Vn = (W(n),WBn) - W(n))(n = 1,2,...) and a,0 be real

numbers such that a2 + j32 = 1. Then by Lemma 8.1 and continuity of W the

set of limit points of the sequence

y/2n log log n jJ n—1

_

I aW(n) + 0(WBn) -W(n)))°°\ y/2n log log n J n=1

is the interval [—l,+l]. This implies that the set of limit points of the sequence

{bnVn} is a subset of the unit disc and the boundary of the unit circle belongs to

this limit set.

Now let V* = (W(n),WBn) - W(n),W{3n) - W{2n)). In the same way as

above one can prove that the set of limit points of {bnV^} is a subset of the unit

ball of J?3 which contains the boundary of the unit sphere. This fact in itself

already implies that the set of limit points of {6nVn} is the unit disc of R2 and

this, in turn, is equivalent to our statement.

Proof of Strassen's Theorem 2. For each u^fiwe have

sup0<

w (x) - w^(x)\ < sup sup \wn(x + s) - wn(x)\,0<z<l

hence by Theorem 7.13 (cf. also Example 3 of Section 7.2)

limsup wn(x) — wlf'(x)\ = d~1'2 a.s.

Consequently we have the Theorem by Lemmas 8.2 and 8.3 where we also use

the fact that Lemma 8.2 guarantees that 5 is closed.

The discreteness of n is inessential in this Theorem. In fact if we define

wt[x)=btW{tx) (i6[0,l],t>0)

then we have

STRASSEN'S THEOREM 3 A964). The net wt(x) is relatively compact in

C@,1) with probability 1 and the set of its limit points is S.

As an application of Strassen's theorem we sketch the proof of Theorem 7.18

in the special case at = at.

At first we mention that Strassen's theorem implies that

\ims\ipbtJi(t,at) = a1^2 a.s.

t—oo

which can be obtained by considering the function

,( \ - I xa~1/2 if °<x<<*,

M*J-jai/2 if a<x<x

in Strassen's class S (cf. also Theorem 7.13 and Example 3 of Section 7.2).The fact that bt is the right normalizing factor for the liminf also follows from

Strassen's theorem. Let

Ca = — liminf btJ\{t, at).t—*oo

In case a = 1 it is well known that C\ = 1 and this can be obtained by consideringthe function f(s) = -s @ < s < 1) in S. Considering the function f(s) = -s

it is also immediate that Ca > a. Theorem 7.18 claims, however, that equalityholds (i.e.Ca = a) if and only if \/a is an integer; in other cases Ca > a.

Now we show that the Strassen's theorem implies that

liminf 6(t)Ji(t, erf) < -Ca = - (B?" t^"

l) . (8.8)*-»oo y r{r + 1) J

Define the function x(s) as follows: if I/a = r (an integer), then let x{s) = -s.

If I/a = r + t, where r is an integer and 0 < x < 1, then split the interval [0,1]into 2r + 1 parts with the points

u2, = ia i = 0,1,2,..., r

= (i + r)a i = 0,1,2, ...,r.

84 CHAPTER 8

Let x(s) be a continuous piecewise linear function starting from 0 (i.e. x[0) = 0)and having slopes

lr(r+l)V/2 .,if u2t < s < u2t+1,r + 1 \ar + 1-t

ar + 1 — tU2i.

It is easily seen that x(s) so defined is in Strassen's class, i.e. x@) = 0, x(s) is

absolutely continuous for 0 < 5 < 1 and Jq x'2(s)ds = 1. Since

x(s + a) - x(s) = -Ca, 0 < s < 1 - a

we have (8.8). Unfortunately we cannot accomplish the proof of Theorem 7.18

by showing that x(s) defined above is extremal within S. In Csaki - Revesz

A979) the proof was completed by some direct probabilistical ideas. The details

are omitted here.

Here we mention a few further applications of Strassen's theorem given byStrassen A964). At first we present the following:

Consequence of Strassen's Theorems 1 and 2. If (p is a continuous func-

functional from C@,1) to R1 then with probability 1 the sequences <p(wn{i)) and

<p(sn(t)) are relatively compact and the sets of limit points coincide with <p[S).Consequently

\ims\ip<p(wn(t)) = \imsuip<p(sn(t)) = sup^>(x) a.s.

n—»oo n-*oo zGS

Applying this corollary to the functional

<p(x) = jT1 x(t)f(t)dt (*€C@,l))

where f(t) @ < t < 1) is a Riemann integrable function with

we obtainr1 r1

limsup / wn(t)f(t)dt = limsup / sn(t)f(t)dtn—*oo JO n—kx> JO

1n f i\

= limsup-V/ - b(n)Si = sup<p(x), (8.9)

and by integration by part we get

Ofi \ 1/2

(F(t))*dt) . (8.10)o /

The above consequence also implies:For.any a > 1 we have

\imsup n-1 (b(n)) J ^l^V\Jo [I — i

in particular

limsupn" 6(n) z-< 1^*1 =3'

a.s.,

n

7t- = 4tt~ a.s.

Remark 2. In order to prove (8.11) we have to prove only that

VVJo A-

This can be done by an elementary but hard calculation.

Similarly we obtain

n)J^—= 2p a.s.

where p is the largest solution of the equation

A - pj^sin (p-^l - pI/2) + cos (A - Pyl*p-') = 0.

A further application given by Strassen is the following. Let 0 < c < 1 and

fi if Si>c(b(i))-\'l

\0 otherwise.

86 CHAPTER 8

Consider the relative frequency qn = n'1 ?"_3 ct. We have

Iimsup7n = 1 -

exp (-4(—- 1)) a.s.

Strassen also notes:

"For c = 1/2 as an example we get the somewhat surprising result that with

probability 1 for infinitely many n the percentage of times i < n when 5, >

l/2Bi'loglogi'I/2 exceeds 99.999 but only for finitely many n exceeds 99.9999."

Finally we mention a very trivial consequence of Strassen's theorem.

THEOREM 8.2 The set

{btm+(xt);0<x<l} [t -> oo)

and the sequence

;0 < x < 1J (n - oo)

are relatively compact in C@, l) with probability 1 and the set of their limit pointsis the set of the nondecreasing elements of S. The analogous statements for m(t)and M(n) are also valid.

8.2 Strassen theorems for increments

As we have already mentioned, Khinchine's LIL is a simple consequence of

Strassen's theorem 1. Here we are interested in getting such a Strassen typegeneralization of Theorem 7.13. At first we mention a trivial consequence of

Theorem 7.13.

Consequence 1. For almost all lj G Q and for all e > 0 there exists a To =

T0(e,u) such that for all T > To there is a corresponding 0 < t = t(u,e,T) <

T —

ar such that

W{t + aT) - W{t) > A - e)G(r,aT))-1 « A - e){2aT logTa^I'2 (8.12)

provided that a? satisfies conditions (i), (ii), (iii) of Theorem 7.13.

Knowing Consequence 1 of Section 8.1 we might pose the following question:does inequality (8.12) imply that W(x) is increasing nearly linearly in (t,t +

a-r)l The answer to this question is positive in the same sense as in the case of

Consequence 1 of Section 8.1.

In order to formulate our more general result introduce the following nota-

notations:

(i) I\T(x) = 7(r,ar)(W(t + xaT) - W(t)) @ < x < 1),

(ii) for all u G Cl define the set VT = Vt(uj) C C@,1) as follows:

Vr = (I\r(z) : 0 < t < T - aT},

(iii) for any A C C@,1) and e > 0 denote ?/(A, e) be the ^-neighbourhood of

A in C@,1) metrics, that is a continuous function a(x) is an element of

U(A,e) if there exists an a(x) G A such that supo^^ | a(x) - a(x) \< e.

Now we present

THEOREM 8.3 (Revesz, 1979). For almost all u ? Q and for all e > 0 there

exists a TQ = To(u,e) such that

U{VT{u),e)DS (8.13)

and

U(S,e)DVT(uj) (8.14)

ifT>T0 provided that ax satisfies conditions (i), (ii), (iii) of Theorem 7.13.

To grasp the meaning of this Theorem let us mention that it says that:

(a) for all T big enough and for all s(x) € S there exists a.0<t<T —

aT such

that TtiT(x)@ < x < 1) will approximate the given s(x),

(b) for all T big enough and for every 0 < t < T -

aT the function I\r(a:)@ <

x < 1) can be approximated by a suitable element s(x) G S.

We have to emphasize that in Theorem 8.3 we assumed all the conditions (i),(ii), (iii) of Theorem 7.13. If we only assume conditions (i) and (ii) then we geta weaker result which contains Strassen's theorem 3 in case at = T.

THEOREM 8.4 (Revesz, 1979). Assume that aT satisfies conditions (i) and

(ii) of Theorem 7.13. Then for almost all u G Q and for all e > 0 there exists a

Tq = T0(e,lj) such that

VT{uj)cU{S,e)

if T > Tq. Further, for any s = s(x) G S, e > 0 and for almost all u> G Q there

exist a T = T(e, w, s) and a 0 < t = t(e, u>, s) < T —

a? such that

sup \VttT{x) -s{x)\ <e.0<z<l

88 CHAPTER 8

Remark 1. The important difference between Theorems 8.3 and 8.4 is the fact

that in Theorem 8.3 we stated that for every T big enough and for every s(x) G S

there exists a 0 < t < T —

ax such that Tt>T(x) approximates the given s(x); while

in Theorem 8.4 we only stated that for every s(x) G S there exists a T (in fact

there exist infinitely many T but not all T are suitable as in Theorem 8.3) and

a.0<t<T— ax such that Ttj(x) approximates the given s(x).In other words if ar is small (condition (iii) holds true), then for every T (big

enough) the random functions Ytj{x) will approximate every element of S as t

runs over the interval [0, T — ay]. However, if ar is large then for any fixed T the

random functions I\x(a:)@ < t < T — ar) will approximate some elements of S

but not all of them; all of them will be approximated when T is also allowed to

vary.

8.3 The rate of convergence in Strassen's

theorems

Property 1. Strassen's theorems 1 and 2 can be reformulated as follows: forany e > 0 and for almost all uj G H there exists an integer no = no(e,u) > 0 such

that

sn(x,uj) G U(S,e) and wn(x,lj) G U(S,e)

if n > no, equivalently there exists a sequence en = en \ 0 such that

sn(x,uj) G U(S,en) and wn(x,uj) G U(S,en) a.s.

for all but finitely many n.

It is natural to ask how can we characterize the possible en sequences in

the above statement. This question was proposed and firstly investigated byBolthausen A978). A better result was given by Grill A987/A), who proved

THEOREM 8.5 Let

^Mn) = (log log n)"*.

Then

sn(x) G U(S,tl)s[n)) and wn(x) G C/(S,^(n)) a.s.

for all but finitely many n if 6 < 2/3; while for 6 > 2/3

sn{x) ? U(S',^«(n)) and ^n(^) ^ U(S,tl>s{n)) i-o. a.s.

Clearly Theorem 8.5 implies Property 1 of Section 8.1 but it does not contain

Property 2 of Section 8.1. As far as Property 2 of Section 8.1 is concerned one

can ask the following question.Let f(x) be an arbitrary element of S. We know that for all e > 0 and for

almost all u> € Q there exists an integer n = n(e,u) resp. n = n(e, u) such that

sup \sn(x) - f[x)\ < e resp. sup \iVn{x) - f(x)\ < e.

0<z<l 0<z<l

Replacing e by en in the above inequalities, they remain true if en | 0 slowlyenough. We ask how such an en can be chosen. This question was raised and

studied by Csaki. He proved

THEOREM 8.6 (Csaki, 1980). For any f(x) € S and c > 0 we have

sup \wn(x) - f(x)\ < c(loglogn)�/2 i.o. a.s.

0<x<\

and

sup \wn{x) - f{x)\ > -A - c)(loglogn) a.s.

o<z 0 then

sup \wn(x) - f(x)\ < —

rrj-—r— i.o. a.s.

o<x<i 4A - aI'2 log log n

and


In case fo[f'(x)Jdx = 1 the best possible rate is available only for piecewise linear

functions. Let f(x) be a continuous piecewise linear function with /@) =0 and

f'(x) = fa if a,_i < x < a, (t = 1,2,... fc)

where a0 = 0 < ax < ... < a*-! < a* = 1. Then we have

90 CHAPTER 8

THEOREM 8.8 (Csaki, 1980). If f[x) is defined as above and f*(f'(x)Jdx =

1 then for any s > 0

sup \wn{x) - /(x)| < ^IZ2-SIZB~1IZ{1 + e)(loglogn)�/3 i.o. a.s.

0<z<l

and

sup \wn{x) - f{x)\ > ^lz2-hlzB-l'z{l - e)(loglogn)�/3 a.s.

0<z<l


Remark 1. Theorems 8.5 resp. 8.8 imply that for any /? < 2/3

W[t) < («Bloglog« + (loglog«I^)I/2 a.s., (8.15)

if t is big enough resp.

W{t) > (t Bloglog« - A + ^(loglogO^V/^1/3^-1/3)) i.o. a.s. (8.16)

(8.15) and (8.16) clearly imply the Khinchine's LIL but they are much weaker

than EFKP LIL of Section 5.2 (cf. Consequence 1 of Section 5.2).

Remark 2. Applying Theorem 8.8 for f(x) = 0 @ < x < 1) we obtain (8.16) as

a special case.

Remark 3. It looks an interesting question to find a common generalization of

the results of Section 8.2 and those of Section 8.3, i.e. to investigate the rate of

convergence in Theorems 8.3 and 8.4. This question was studied by Goodman

and Kuelbs A988).

Remark 4. Csaki (personal communication) recently found the generalizationof Theorem 8.8 in the case when f(x) is a quadratic function (and of course

1

8.4 A theorem of Wichura

We have seen that Strassen's theorem is a natural generalization of Khinchine's

LIL. Wichura proposed to find a similar (Strassen type) generalization of the

Other LIL (cf. E.9)). He proved

WICHURA'S THEOREM (Wichura, 1977, Mueller, 1983). Consider the

sequence

yp|M ) H||; 0< u<

J \ n )

(n = 1,2,...) and the net

tit(u) = | 1logl�" 1 sup \W(tx)\; 0 (t -> oo).

^V l J *<« J

Let ? 6c t/ic set o/ nondecreasing, nonnegative functions g on [0, l] satisfying

i:Then with probability 1, the set of limit points of sn(u) resp. iOj(u) in the weak

topology, as n /* oo resp. t / oo is Q.

Remark 1. In order to see that this theorem implies the Other LIL we onlyhave to prove that g(l) > y| for any g(-) € Q.

Chapter 9

Distribution of the local time

9.1 Exact distributions

Let

Px(fc) =min{n:5n = k} {k = 1,2,.. .)•

Then

Hence by Theorem 2.4 we obtain

THEOREM 9.1

(fc = l,2,...,n = l,2,...).

j=°x[~2~yEspecially

Consequently

1) = 2m + 1} = 2-2~-1(m + I)�^ (9.1)

Note that /?iBA: + 1) takes only odd, /?iBA:) takes only even numbers.

93


94 CHAPTER 9

THEOREM 9.2 Let p0 = 0 and pk = min{j : j > pk-i, S}¦ = 0}(/c = 1,2,...).Then px, p2

—

p\,pz—

P2, ¦ • • I5 ° sequence of x.i.d.r.v's wxth

¦p/_ — Ot\ — O-2*+1i--1 I 1 ft- 1 O ^ (Q O\^ J \ Ic I /

\ /

and

> 2n) = 2'2n[ n) = P{52n = 0}.

Proof. The statement that Pi,P2~ Pi,Pz~ P2,- ¦ • are i.i.d.r.v.'s taking only even

values is trivial. Hence we prove (9.2) only.

J>{Pl = 2k} = ip{Pl =2k\Xi = +1} + \p{pi = 2k\Xx = -1}.2 &

Clearly

l= 2k | Xx = +1} = PI/?! = 2A: | Xx = -1} = P{pi(l) =2k- 1}. (9.3)

Hence by (9.1) we have (9.2). The second statement of Theorem 9.2 is a

simple consequence of (9.2).

Remark 1. A simple calculation gives

f = 2k) =

Hence the particle returns to the origin with probability 1, i.e. we obtained a

new proof of Polya Recurrence Theorem of Section 3.1. However, observe that

i.e. the expectation of the waiting time of the recurrence is infinite.

Consider a random walk {5*; A: = 0,1,2,...} and observe how many times a

given x(x = 0, ±1,±2,...) was visited during the first n steps, i.e. observe

?{x,n)=#{k:0<k<n,Sk = x}.

The process f (a:, n) (of two variables) is the local time of the random walk {Sk}.

DISTRIBUTION OF THE LOCAL TIME 95

THEOREM 9.3 For any k = 0,1,2,..., n; n = 1,2,... we /iave

n) = k}= P{?@,2n + 1) = *} = 2

Equivalently

*-i (on - A> 2n} = P{e(O, 2n) < A:} = 2�" ^ 2'

J.

i=o V n/

Proof. (9.3) implies that the distribution of pi is identical with that of pi[l) +1.

It follows that the distribution of pk— k is identical to that of Pi(k), i.e.

-k>n} = P{Pl{k) >n}= P{Mn+

Further, we have

P{f@,2n) =k} = P{pk < 2n,pk+1 > 2n)

which implies the Theorem.

Applying Theorems 9.1 and 9.3 we can get the distribution of f (a:, n). In fact

we have

THEOREM 9.4 Let x > 0. Then for any k = 1,2,...

,n) = k}= j;*P{e(x,n) = k | Pl(x)j=x

J=X

r

and for k = 0

l=n+l*

\ o '

96 CHAPTER 9

Theorem 9.3 gave the distribution of the number of zeros in the path So, Si,...,

S^n+i- We ask also about the distribution of the number of those zeros where

the path changes its sign. Let

0(n) = #{A; : 1 < k < n,Sk.xSk+i < 0}

be the number of crossings (sign changes). Then as a trivial consequence of

Theorem 9.3 we obtain

THEOREM 9.5

P{0Bn + !)=*} = ? P{0Bn + l) = k | ?@,2n) = j}P{?@,2n) = j}j=k

)() = (j?k \k) 2>\ n ) 22n~> 22n \n +

= 2P{52n+1 =2k + 1}.

Proof. Observe that P{5jfe_i5jfe+1 < 0 | 5* = 0} = 1/2.

THEOREM 9.6

P{ max Si > n} = Bn)� (n=l,2,...).

Proof. It is a trivial consequence of Lemma 3.1.

THEOREM 9.7 For any k = ±1, ±2,... and 1 = 0,1,2,... we have

(9.4)

)' (9-5)

= l, E(?(A:,p1)-lJ=4A:-2. (9.6)

Proof. Without loss of generality we may assume that k > 0. Then

{?(*,*) = o} = {Xi = -1} u {Xi = i,s2 # k,s3 # k,...,spi_i # k}.

Hence by Lemma 3.1

1 1 A: — 1 2A;-1

and (9.4) is proved.For any x = ±1, ±2,... define

po[x) = 0,

Pl{x) =inf{l:l>0,Sl = x},

pi+1{x) = inf{/ : / > Pi{x), S, = x} (i = 0,1,2,...).

Then in case m > 0 we have

{t[k,Pi) = m} = {0 < Pl(k) < P2{k) <...< pm[k) <Pl

Hence, again by Lemma 3.1

P{€(*,*) = -I =

-2l [E { j ) B) (j- A \~

\2k)

2k

2k

and (9.5) is also proved.(9.6) is a simple consequence of (9.4) and (9.5).Define the r.v.'s ftn as follows: f2n A = 1,2,...) is the number of those terms

of the sequence 5l5 S2,..., ^n which are positive or which are equal to 0 but the

preceding term of which is positive. Then ftn takes on only even numbers and

its distribution is described by

THEOREM 9.8

B*)(^lfJ�B (* = 0,l,.-.,»). (9-7)

Proof. (Renyi, 1970/A). Clearly

{f2n = 0} = {M2+n = 0}.

Hence by Theorem 2.4

n\2~2n,

i.e. (9.7) holds for k = 0. It is also easy to see that (9.7) holds for n = 1, k = 0,1.Now we use induction on n. Suppose that (9.7) is true for n < N—l and consider

98 CHAPTER 9

= 2k} for 1 < k < N - 1. If <;2N = 2k and 1 < k < N - 1 then the

sequence SX,S2, ¦ ¦ ¦ ,S2N has to contain both positive and negative terms, and

thus it contains at least one term equal to 0. Let pi = 21. Then either Sn > 0

for n < 21 and S2i = 0 or Sn < 0 for n < 21 and 52j = 0. Both possibilities have

the probability

(cf. (9.2)).Now if Sn > 0 for n < 21 and S2l = 0, further if $2N = 2k, then among the

numbers S2l+i,... ,S2N there are 2k — 21 positive ones or zeros preceded by a

positive term, while in case Sn < 0 for n < 21, S2l = 0 and $2N = 2k, the number

of such terms is 2k. Hence

W = 2/}P{f2*_2, = 2k- 21}

+ \fl p(^i = 2OPU2*-* = 2A;}1

1=1

and we obtain (9.7) by an elementary calculation.

It is worthwhile to mention that the distribution of the location of the last

zero up to 2n, i.e. the distribution of

*Bn) = max{Jk : 0 < k < n, S2k = 0}

agrees with the distribution of $2n. In fact we have

THEOREM 9.9

Proof. Clearly by Theorem 9.2

P{#Bn) = 2k} = P{S2k = 0}P{Pl > 2n - 2k} = P{52Jfe = 0}P{52n_2Jfe = 0}.

Hence the Theorem.

The distribution of the location of the maximum also agrees with those of fn

and \&(n). In fact we have

THEOREM 9.10 Let

fi+{n) = inf{k:0<k<n for which S{k) = M+(n)}.

Then

P{M+Bn)=A;} =

(B[k/2]\Bn-2[k/2\\[m){n-[k/2]J»-2n A; = 0.

n

Proof. Clearly the number of paths for which

is equal to

Hence

P{M+Bn)

the number of

} = P{S0 -

= P{50 -

paths

{Si >

<Sk,l

for which

0,52 >0,

?! < Sk,..

h<sk,..

Sk, Sk+1 < Sk,..., S2n < Sk}

?! > 0, S2 > 0,..., Sk > 0}'P{S1 > 0,..., 52n_jfe > 0}.

Then we obtain Theorem 9.10 by Theorem 2.4.

9.2 Limit distributions

Applying the above given exact distributions and the Stirling formula we obtain

THEOREM 9.11

^ / y« *\ 1 1*1.

v e av, [\).a)*-

(9.9)

where \&k\ < 1,

(9.10)

1Oo CHAPTER 9

HmPJn-^eCO.n) < x\ = lim P {n-l<2t[z,n) < x}n—>oo I¦» \ / j n—>oo «. '

e'u2/2du (z = ±l,±2,...), (9.11)

{ n J n~>0° I n J r»->0° I n

2= -arcsinV* @ < x < 1). (9.12)

7T

Remark 1. (9.12) is called arcsine law. It is worthwhile to mention that by(9.12) we obtain

limp(o,45< - < 0,55} = 0,063769...n—00 ^ n )

and

lim P (^ < 0, l} = lim P (^ > 0,9} = 0,204833...

The exact distribution (9.7) of ftn als° implies that the most improbable value

of ftn is n and the most probable values are 0 and 2n. In other words with a bigprobability the particle spends a long time on the left-hand side of the line and

only a short time on the right-hand side or conversely but it is very unlikely that

it spends the same (or nearly the same) time on the positive and on the negativeside.

9.3 Definition and distribution of the

local time of a Wiener process

It is easy to see that the number of the time points before any given T, where a

Wiener process W is equal to a given x, is 0 or 00 a.s., i.e. for any T > 0 and

any real x

#{*:0< t <T,W(t) =x) = 0oroo a.s.

Hence if we want to characterize the amount of time till T which the Wiener

process spends in x (or nearby) then we have to find a more adequate definition

than the number of time points. P. Levy proposed the following idea.

Let H(A,t) be the occupation time of the Borel set A C R1 by W(-) in the

interval @,t), formally

H{A,t) = \{s:s<t,W{s)

where A is the Lebesgue measure.

For any fixed t > 0 and for almost all u; € Cl the occupation time H(A,t)is a measure on the Borel sets of the real line. Trotter A958) proved that this

measure is absolutely continuous with respect to the Lebesgue measure and its

Radon - Nikodym derivate rj(x,t) is continuous in (x,t). The stochastic process

ri(x,t) is called the local time of W. (It characterizes the amount of time that

the Wiener process W spends till t is "near" to the point x.) Our first aim is to

evaluate the distribution of the r.v. 77@, t).In fact we prove

THEOREM 9.12 For any x > 0 and t > 0

9 rz

(9.13)1

7T Jo

(N) (N)Proof. For any N = 1,2,... define the sequence 0 < T\ = t{

'< r2 = r2v

;< ...

as follows:

Tx = M{t:t>0,\W[t)\ = N-1},r2 = inf{* :t>ru \W(t) - W{Tl)\ = N-1},

ri+1 = inf{* : * > r.-, \W{t) - W{n)\ = N'1},

(cf. Skorohod embedding scheme, Section 6.3) and let

siN)=W{rk) (A; = 1,2,...),

u = u^N) = max{i : r, < 1}.

Note that Ti,t2—

Ti,... is a sequence of i.i.d.r.v.'s with

Et"! = N~2 and ErJ < 00 (9.14)

(cf. F.1) and Theorem 6.3).The interval (rj,rl+i) will be called type (a,6) (a = jiV�,^— a\ = N-1,j =

0,1,2,...) if \W(Ti)\ = a and |W(ri+i)| = 6. The infinite random set of those

j's for which (r,-,Tj+1) is an interval of type (a, 6) will be denoted by I^(a,b) =

I(a, 6). It is clear that |W(*)| can be smaller than JV� if t is an element of an

interval of type (O,^�), {N-\0) or (iV�^^�). Let

A = AW = {i:0<i<u,ie /(O,^�) U

102 CHAPTER 9

Then by the law of large numbers and (9.14)

7Ti) -»1 a.s. (JV-oo). (9.15)

(In fact (9.15) can be obtained using the "Method of high moments" of Section

4.2; to obtain it by "Gap method" seems to be hard.)Studying the local time of W(-) in intervals of type (N~l,2N~1), we obtain

5>(O,rl+1)-r7(O,r,)) = O a.s. (9.16)

for any N = 1,2,... where

B = BN = {i : 0 e0 if t e U.esK^.+i)-)Hence

1,@,1) = Jim ^ ?(rI+1 - r.-) a.s. (9.18)N^°° 2

i€A

Then, taking into account that limjv_oo N~2uN = 1 a.s., (9.11), (9.15) and

(9.18) combined imply that

Jl[X (9.19)

and Theorem 9.12 follows from (9.19) and from the simple transformation: for

any T > 0

{r,[x,tT),x e R\0 <t<l} = {Tll2r)[xT-ll2,t),x eR\0<t< l}. (9.20)

Theorem 9.12 clearly implies that for any x € R1 we have

Levy A948) also proved

THEOREM 9.13 For any xe R1, T > 0 and u > 0 we have

To evaluate the distribution of 77 (t) = s\ip_O0<x<O0rj(x,t) is much harder. This

was done by Csaki and Foldes A986). They proved

THEOREM 9.14

where 0 < j\ < j2 < ... are the positive zeros (roots) of the Bessel function

Jo(x) = Io(ix) and for any k = 1,2,...

4ak =

. 2 • »

sm j*

6, = 4 I-] ^

Jk7rJ0(A;7r) \J0{kn)) ]'

Ji(A;7r)

1

i

Furthermore

i / 2j,2\<z^«aiexp f as z -¦ 0.

Remark 1. The proof of Theorem 9.14 is based on a result of Borodin A982),who evaluated the Laplace transform of the distribution of t~ll2r){i).

Chapter 10

Local time and Invariance Principle

10.1 An Invariance PrincipleThe main result of this Section claims that the local time f(:r,n) of a random

walk can be approximated by the local time ri(x, n) of a Wiener process uniformlyin x as n —¦ oo. In fact we have

THEOREM 10.1 (Revesz, 1981). Let {W(t),t > 0} be a Wiener process de-

defined on a probability space {fl, J,P}. Then on the same probability space Q one

can define a sequence Xi, X2,... of i.i.d.r.v. 's with P(X, = 1) = P(X, = —1) =

1/2 such that

lim rT1/4~esup|f(:r,n) - r)(x,n)\ =0 a.s. A0.1)n—>oo

x

for any e > 0 where the sup is taken over all integers, r\ is the local time of W

and ? is the local time of Sn = Xi + X2 H h Xn.

For the sake of simplicity, instead of A0.1) we prove only

lim n-<~e|f@,n) - ri@,n)\ =0 a.s. A0.2)n—>oo

for any e > 0. The proof of A0.1) does not require any new idea. Only a more

tiresome calculation is needed.

Proof of A0.2). Define the r.v.'s r0 = 0 < n < t2 < ... just like in Section

6.3. Further let 1 < fix < \i2 < ... be the time-points where the random walk

{Sk} = {W(rk)} visits 0, i.e. let

Mi = min{Jk : k > 0, W{rk) = Sk = 0},M2 = min{A;: k > fiuW[Tk) = Sk = 0},

105

106 CHAPTER 10

Mn = min{A; : k > Hn-i,W{rk) = Sk = 0},

Then

f@,n) = max{A; : \ik < n}

and€@.*)

The proof of Theorem 9.12 implies

Hence

() a.s. (k - oo).

F.1) easily implies that

r* = k + o (it'+e) a.s.

Then A0.2) easily follows from

f(O,Jt) =o(k*+e) a.s. (ife-»oo) A0.3)

and

sup {p @,j + k^') - v{O,J)) = o (k*+°) a.s. {k -> oo). A0.4)

A0.3) and A0.4) can be easily proved. Their proofs are omitted here because

more general results will be given in Chapter 11.

Remark 1. It turns out that the rate of convergence in Theorem 10.1 is nearlythe best possible. In fact Remark 3 of Section 11.5 implies that if a Wiener

process W(-) and a random walk {Sn} are defined on the same probability space

then

limsupn~1/'4sup|^(x,n) - ?7(x,n)| > 0 a.s. A0.5)n—>oo x

However, the answer to the following question is unknown. Assume that a Wiener

process and a random walk are defined on the same probability space and

lim n~a I f@, n) -

r\ @, n) \ = 0 a.s.n—*oo

What can be said about a?

LOCAL TIME AND INVARIANCE PRINCIPLE 107

Remark 2. It can be also proved that in Theorem 10.1 the random walk Sn and

the Wiener process W(t) can be constructed so that besides A0.1)

\Sn-W{n)\ =O(logn) a.s.

Remark 3. A trivial consequence of Theorem 10.1 is

Km'P{n-1t2?{n) < z) =P{r?(l) < z) A0.6)

for any z > 0 where f(n) = maxx ?(x,n) (cf. Theorem 9.14).

10.2 A theorem of Levy

Theorem 10.1 tells us that the properties of the process ?(x,n) are the same

(or more or less the same) as those of rj(x,n). In other words studying the

behaviour of one of the processes ?(x,n),ri(x,n) we can automatically claim

that the behaviour of the other process is the same. The main results of the

present section tell us that the properties of f@, n) resp. rj@,n) are the same

as those of M+(n) resp. m+(n). Hence the theorems proved for M+(n) resp.

m+(n) will be inherited by f@, n) resp. rj@,n).Let

y{t) =m+{t)-W{t) (t>0)and

Y[n) =M+{n)-S{n) (n = 0,1,2,...).Then a celebrated result of P. Levy reads as follows (see for example, KnightA981), Theorem 5.3.7):

THEOREM 10.2 We have

{y{t),m+{t);t > 0}l{\W{t)\,r,{0,t);t > 0},

i.e. the finite dimensional distributions of the vector valued process {y(t), m+(t);t > 0} are equal to the corresponding distributions of {\W(t)\,ri(Q,t);t > 0}.

In order to see the importance of this theorem, we mention that applyingthe LIL of Khinchine (Section 4.4, see also Theorem 6.2) for m+(t) as a trivial

consequence of Theorem 10.2 we obtain

Consequence 1.

limsup /^ =1 a.s. A0.7)t-oo B* log log*I/2

108 CHAPTER 10

In fact the Levy classes can be also obtained for rj(Q,t).Applying Theorem 10.1, Consequence 1 in turn implies

Consequence 2.

lfi =1 a.s. A0.8)msupplfivn—oo y/2n log log fl

Remark 1. A0.7) was proved (directly) by Kesten A965). A0.8) is due to

Chung and Hunt A949).A natural question arises: what is the analogue of Theorem 10.2 in the case

of a random walk? In fact we ask: does Theorem 10.2 remain true if we re-

replace W(t),y(t),m+{t),rl{0,t) by S{n),Y{n),M+{n) and f@,n) respectively?The answer to this question is negative, which can be seen by comparing the

distributions of f@,2n) and M+[2n) (cf. Theorems 2.4 and 9.3).In spite of this disappointing fact we prove that Theorem 10.2 is "nearly true"

for random walks. In fact we have

THEOREM 10.3 (Csaki - Revesz, 1983). Let Xi,X2,... be a sequence ofi.i.d.r.v.'s with P(Xi = 1) = P(Xi = —1) = 1/2 defined on a probability space

{fi, J,P}. Then one can define a sequence Xi, X2,... of i.i.d.r.v. 's on the same

probability space {Q, J,P} such that P(XX = 1) = P(XX = -1) = 1/2 and forany e > 0

n-'\Y{n) -\S[n)\ | -* 0 a.s.

and

n-1/4"e|M+(n) -?@,n)| -* 0 a.s.,

where

M+(n) = max S(A:), 5@) = 0, S{n) = Y^Xk (n = 1,2,...),°-*-n

*=i

Y{n) =M+{n) - S[n).

Remark 2. This theorem is a bit stronger than that of Csaki - Revesz A983).The proof is presented below.

Remark 3. Consequence 2 can also be obtained by applying the LIL of Khin-

chine (cf. Section 4.4) for M+ and Theorem 10.3.

Theorem 10.3 tells us that the vector (|5(n)|, f@,n)) can be approximatedby the vector (Y(n),M+(n)) in order n1/4�"*. Unfortunately we do not know

what the best possible rate here is. However, we can show that by considering


the number of crossings €>(n) instead of the number of roots f@, n), better rates

can be achieved than that of Theorem 10.3. Let

e(n) = #{Jt : 1 < k < n, S{k - l)S(k + 1) < 0}

be the number of crossings. Then we have

THEOREM 10.4 (Csaki - Revesz, 1983 and Simons, 1983). Let XUX2,...be a sequence of i.i.d.r.v.'s with P(Xi = 1) = P(Xi = -l) = 1/2 definedon a probability space {fl, J,P}. Then one can define a sequence Xi,X2,...of i.i.d.r.v.'s on the same probability space {fi, J,P} such that P(Xi = 1) =

P(Xi = -1) = 1/2 and

|M+(n)-20(n)| < 1, A0.9)

|F(n)-|5(n)||<2, A0.10)

for any n = 1,2,... where

M+{n) = max 5 (Jfc), 5@) = 0, S(n) = J^ Xk (n = l,2,...),

Y{n) =M+{n)-S{n).

Proof. Let

tx = min{i : i > 0, S{i - lM(i + 1) < 0},t2 = min{i : i > r,, S{i - l)S{i + 1) < 0},

= min{t : i > n,S{i - l)S{i + 1) < 0},

andif

xxxj+l if

if TJ + 1 < J <

This transformation was given by Csaki and Vincze A961). The following lemma

is clearly true.

HO CHAPTER 10

LEMMA 10.1

(i) X\, X2,... is a sequence of i.i.d.r.v. 's with

00

S{k)-

(Hi) 26(r,) =2/ = 5(r«) =M+(r,), / = 1,2,....

(iv) For any n < n < rJ+1 we have &(n) =1,21 < M+(n) < 2/ +1, consequently

0<M+(n)-26(n) < 1.

(v)J 1)| if r, +

if k =

therefore

?{k)=M+{k)-~S{k) < \S{k + l)\ < \S{k)\ + l

and

Y{k) = M+{k) - 5(ik) > \S{k + 1)| - 1 > \S{k)\ - 2.

This proves Theorem 10.4.

Proof of Theorem 10.3. Clearly we have

n-i/4-'|f@,n) - 20(n)| -* 0 a.s. A0.11)

Hence we obtain Theorem 10.3 as a trivial consequence of Theorem 10.4.

Applying the Invariance Principle 1 (cf. Section 6.3), Theorems 10.2 and 10.4

as well as A0.11) we easily obtain

Consequence 3. (Csaki - Revesz, 1983). On a rich enough probability space

{fl, J,P} one can define a Wiener process {W(t);t > 0} and a sequence Xu X2,...

of i.i.d.r.v.'s with PpG = 1) = T{X1 = -1) = 1/2 such that

\S{n)-W{n)\ =O(logn) a.s.,

|2e(n)-»7(O,n)| =O(logn) a.s. A0.13)

and for any e > 0

|e(O,n)-17@,11I = o(n1/*+«). A0.14)

Remark 4. Hence we obtain a new proof of Theorem 10.1 when only a fixed x

is considered.

Remark 5. Having Theorem 10.3, Theorem 10.2 can be easily deduced (cf.Csaki - Revesz 1983, Simons 1983).

Question. Is it possible to define two random walks {S^} and {S^} on a

probability space such that

rra|fW@,n)-2eB)(n)| -* 0 a.s.

for some 0 < a < 1/4 (cf. A0.14)) where fA)@,n) is the local time of S^ and

€>B) is the number of crossings of 5^? If a positive answer can be obtained,then in A0.11) a better rate can be also obtained. However, if the answer to

this question is negative, then A0.11) also gives the best possible rate (exceptthat perhaps ne can be replaced by some logn power). Hence this question is

equivalent to the question of Remark 1 of Section 10.1.

Now we formulate another trivial consequence of Theorem 10.2.

Consequence 4. Let

7{T)= max (W[u)-W[v))V ' 0<u<v<TX

be the maximal fall of W(-) in [0,T|. Then

'2ir n

n *€ LLC(J(T)),

\ 8 log logn/

P {T-X'2J{T) <x}= G{x) = H{x)where G(-) and #(•) are defined in Theorem 2.13.

Proof. Observe that J{T) = maxo<t<r y[t) and apply Theorems 10.2, 2.13, the

LIL of Khinchine (Section 4.4) and the Other LIL E.9).It is also interesting to study the properties of J{T) for those T"s for which

W (T) is very big. In fact we prove

112 CHAPTER 10

THEOREM 10.5 Let C\ and C2 be two positive constants. Then there exists

a sequence 0 < tx = ?i(ur, Ci, C2) < t2 = ^(^J Ci> C2) < ... such that

' '

\loglog*n

if

1/2

and W{tn) > CMtn))'

7T2

The proof of the above theorem is based on the following:

THEOREM 10.6 (Mogul'skii, 1979).

limuMogP ( sup \W{t)-tW{T)\ < uTl'2\ = ~.

"->o ^o<t<r J 8

Proof of Theorem 10.5. Observe that the conditional distribution of

{W{t), 0<t<T} given W[T) = CF(r))�

is equal to the distribution of

{BT{t) + Ct{Tb{T)Y\ 0<t<T}.

Further, the maximal fall of BT[t) + Ct{Tb{T))~l is less than or equal to

2maxo<t<r |Br(OI- Hence we obtain

-1

f-exp (-ir^rloglogT) exp("Ci2 loglogT) -

where BT(t) = W(t) - tW[T) @ < t < T) and the proof follows by the usual

way (cf. e.g. the proof of the LIL of Khinchine, Section 4.4).

Chapter 11

Strong theorems of the local time

11.1 Strong theorems for ?(z,n) and

The Recurrence Theorem (cf. Section 3.1) clearly implies that for any x =

0,±l,±2,...lim i(x,n) = 00 a.s. (HI)

n—»oov '

In order to get the rate of convergence in A1.1) it is enough to observe that byTheorem 10.3 the limit behaviour of f@, n) (and consequently that of ?(x, n)) is

the same as that of M+. Hence by the EFKP LIL (cf. Section 5.2) and by the

Theorem of Hirsch (cf. Section 5.3) we obtain

THEOREM 11.1 The nondecreasing function

if and only if

The nonincreasing function

if and only if

n=lU

where x is an arbitrary fixed integer.

Having Theorem 10.2 (instead of Theorem 10.3) and Theorem 6.2 or applyingTheorem 11.1 and Theorem 10.1 we obtain

113


114 CHAPTER 11

THEOREM 11.2 Theorem 11.1 remains true if we replace ?(•,•) by v(-,-)-

Remark 1. Theorem 11.1 was proved originally by Chung and Hunt A949).Theorem 11.2 is due to Kesten A965).

The study of ?(n) is much harder than that of ?(x,n). However, havingTheorem 9.14 (cf. also A0.6)) one can prove

THEOREM 11.3 (Kesten, 1965, Csaki - Foldes, 1986).

Iimsup6(n)f(n) = lim sup 6(?) 77B) = 1 a.s. (H-2)t—oo

liminfn-1/2(loglognI/2e(n) = liminf r1/2(loglog*I/2r?Mn—»oo t—»oo

= 1 = 21/2ji as- A1.3)

where j\ is the first positive root of the Bessel function Jo(x).

Remark 2. A1.2) is due to Kesten A965). A1.3) is also due to Kesten without

obtaining the exact value of 7.

The result of Csaki and Foldes A986) is much stronger than A1.3). In fact

they proved:

THEOREM 11.4 Let u(t) > 0 be a nonincreasing function such that

oo u(t) = Q,u{t)tll2 is nondecreasing and lim^oo u{i)tll2 = 00. Then

u{i) e LLC (t-lf2Ti(t)) and u{n) € LLC (n�/^(n))if and only if

(Remark 3. The proof of Theorem 11.4 is based on Theorem 9.14.

The upper classes of rj(t) and those of f(n) were also described by Csaki

A989). He proved

THEOREM 11.5 Let a(t) > 0 (t > 1) be a nondecreasing function. Then

a(t) E UUC (r1/2»7(O) and a{n) ? UUC (fTif and only if

a3(t) ( a2f°°a3(t) ( a2(t)\ ,

L -rexp -41 r<o°-

STRONG THEOREMS OF THE LOCAL TIME 115

Since ?(Q,pn) = n, i.e. pn is the inverse function of ?@, n), by Theorem 11.1

we can also obtain the Levy classes of pn. Here we present only the simplestconsequence.

THEOREM 11.6 For any e > 0 we have

n2(lognJ+e€UUC(pn),n2(lognJ-e€ULC(pn),

11.2 Increments of rj(x,t)First we give the analogue of Theorem 7.13.

THEOREM 11.7 (Csaki - Csorgo - Foldes - Revesz, 1983). Let at{t > 0) be a

nondecreasing function of t for which

CO 0 < at < t,

(ii) t/at is nondecreasing.

Then

sup (r)(x,s + at) — rf(x,s))t-»oo

= limsup 6t(r)(x,t) — r}{x,t — at)) = 1 a.s.

If we have also

(Hi)

t—»oo

then

lim(log(«/at))(loglog«)l

= oo—»oo

lim 6t sup (r)(x,s + at) — r)(x,s)) = 1 a.s.t—oo o<*<t-a,

for any fixed x € R1 where 6t = a;1/2(log(t/at) + 2 log log t)~ll2.

By Theorem 10.2 as a trivial consequence of the above Theorem we obtain

THEOREM 11.8 Theorem 11.7 remains true replacing r)(x,t) by m+(t).

116 CHAPTER 11

Remark 1. Clearly

sup {m+{s + at)-m+{s))< sup {W{s + at) - W{s)). A1.4)

Comparing 6t and it of Theorem 7.13 we obtain in A1.4) that for a sequence

t = tn | oo we may have strict inequality whenever (iii) does not hold true.

The investigation of the largest possible increment in t when x is also varyingseems to be also interesting. We obtained

THEOREM 11.9 (Csaki - Csorgo - Foldes - Revesz, 1983). Let at{t > 0) be

a nondecreasing function oft satisfying conditions (i) and (ii) of Theorem 11.7.

Then

limsup^t sup sup (rf(x,s + at) - ri(x,s)) = 1 a.s.

t-»oo xCR1 0<s<t-at

If we also assume that (iii) of Theorem 11.7 holds then

lim^sup sup [rf(x, s + at) — rj(x,s)) = 1 a.s.'—°° 0<<

To find the analogue of Theorem 7.20 seems to be much more delicate. At

first we ask about the length of the longest zero-free interval. Let

r(t) = sup{a : for which 30 < s < t - a such that r)(Q,s + a) - r)(Q, s) = 0}

be the length of the longest zero-free interval. Then we have

THEOREM 11.10 (Chung - Erdos, 1952). Let f(x) be a nondecreasing func-function for which l\mx-.O0f(x) = oo,x/f(x) is nondecreasing and limz—oo x/f(x) =

oo. Then

' t1" m)€ uuc(r(())

if and only if

-

Remark 2. Originally this theorem was formulated for random walk instead of

Wiener process.

Example 1. Since L{f) < oo if f(x) = (logxJ+e(er > 0) and L{f) = oo if

f(x) = (logxJ, we obtain

( W)s uuc(r(())

and

111 - (ii?)€ ULC<r"»-

or equivalently

lim inf inf G7@,5 + at) - 77@,5)) = lim infG7@, t) - 77@, t- at)) > 0t—+oo O^s^t — &t t—*oo

and

lim inf inf G7@,5 + at) - 77@,5)) = lim infG7@,*) - 77@, t- at)) = 0

This example shows that the study of the lim inf properties of info<s<t-a, (^ @,5 +

at) — 7?@,5)) (i.e. the analogue of Theorem 7.20) is interesting only if at >

t(l - (log*)�). This question was studied by Csaki and Foldes A986). Theyproved

THEOREM 11.11 Let fi(t) = t^-at)'1 be a nondecreasing function for which

t/fi(t) is also nondecreasing and limt—oot/fi{t) = cx>,\imt-.oo fi(t) = 00. Fur-

Further, let f2(t) be a nonincreasing function for which lim^oo f2{t) = O,t1/2f2(t) is

nondecreasing and limt_>Oo*1^2/2@ = °°- Then

(t) e LUC { inf G7 @,5 + at) - 77 @,5))}if

L{h) = ao or V (/,) = 00,

and

t^fiit) e LLC {inf G7@,5 + a,) - 77@,5))}if

L{h) < 00 and L*(f2) < 00,

where

118 CHAPTER 11

Example 2. Let at = t[l - (log*)~2~')(e > 0). Then fx{t) = (log*J+e and

L{fi) < oo. Since

_!_«, f = oo if 6 = 0,

we obtain

Remark 3. By Theorems 10.1, 10.2 and 10.3, we find that the statement of

Example 2 remains true replacing 77@,*) by m+(t) or M+(n). (Compare this

result with the Theorem of Hirsch of Section 5.3.)Finally we mention the following analogue of Theorem 11.9 (cf. also Theorem

11.3).

THEOREM 11.12 (Csaki - Foldes, 1986). Let at(t > 0) be a nondecreasingfunction oft satisfying conditions (i) and (ii) of Theorem 11.7. Then

liminft?tQ(*) = 1 a.s.t—»oo

where

Q{t) = inf

and

_ /log(t/q.) + loglogt'\'/'*[ )

If we also assume that (iii) of Theorem 11.7 holds then

lim &tQ{t) = 1 a.s.t»oot—»oo

Remark 4. In case at = t we obtain A1.3) as a special case of Theorem 11.12.

Remark 5. The study of the increments of r}(x,t) in x or in both variables looks

a challenging question.

11.3 Increments of

In Section 10.1 we have seen that the strong theorems valid for r}(x, t) resp. rj(t)remain valid for f(z, n) resp. f(n) due to the Invariance Principle (Theorem

10.1). In Section 7.3 we have seen that the strong theorems proved for the

increments of a Wiener process remain valid for those of a random walk if an ;»

logn resp. an » (lognK depending on what kind of theorems we are talkingabout. This latter fact is due to the Invariance Principle 1 (Section 6.3) and

especially the rate O(logn) in it. Since the rate in Theorem 10.1 is much worse

(it is ©(n1/4"*"') only) we can only claim (as a consequence of the Invariance

Principle) that the results of Section 11.2 remain valid for ?(x, n) (instead of

rj(x,t)) if an > n1/2�"'. The case an < n1/2�"' requires a separate study. This

was done by Csaki and Foldes A984/C). They proved that Theorem 11.7 remain

valid for ?(x, n) if an » logn. In fact they proved the following two theorems:

THEOREM 11.13 Let 0 < an < n(n = 1,2,...) be an integer valued nonde-

creasing sequence. Assume that an/n is nonincreasing and

Then

lim -—— = oo.n—oo log n

Iimsup5n sup {?{x, k + an) — ?{x, k)) = 1 a.s.

n—oo 0<k<n-an

If we also have

log(n/an)lim -—-r1—- = oo

n—oo log log n

then

lim 6n sup {?{x,k + an) — ?(x,k)) = 1 a.s.n^°°

0<k<n-an

for any fixed x 6 Z1 where

6n = a'1'2 (log(nO + 21oglogn)�/2.THEOREM 11.14 Let c> 0. Then for any fixed x € Z1

f(z,/c + [clogn]) - ?{x,k)lim max r— : = a[c) a.s.

n—ooo<*<n-[<;logn] [C log n]

where a(c) = 1/2 if c < (Iog2)-1 and the only solution of the equation

- = A - 2a) log(l - 2a) - 2A - a) log(l - a)C

ifc>

120 CHAPTER 11

Remark 1. The above theorem suggests the conjecture:

=[|] a.S.

for all but finitely many n provided that an — o(logn).Since the Invariance Principle 1 of Section 6.3 is valid with the rate O(logn)

Theorem 11.8 implies

THEOREM 11.15 Theorem 11.13 remains true replacing ?{x,n) by M+(n).

The analogue of Theorem 11.9 for f(z, n) is unknown except if an > n

The analogues of Theorems 11.10 and 11.11 can be obtained by the Invariance

Principle for ?(x, n).

11.4 Strassen type theorems

Let

xt) @<x<l,t>0)

and

Un(x)=bn[t[0,k)+n(x-±)(Z@,k + l)-Z@,k))) if ?<*<^(k = 0,1,2,..., n — 1; n = 1,2,...). We intend to characterize the limit points of

the sequence Un(x) and those of ut(x). Since Un(x)@ < x < 1) for any fixed n is

a nondecreasing function, its limit points must also be nondecreasing.

Definition. Let Sm C S be the set of nondecreasing elements of S (cf. Notations

to the Strassen type theorems).Then we formulate

THEOREM 11.16 (Csaki - Revesz, 1983). The sequence {Un(x);0 < x < 1}and the net {ut(x);0 < x < 1} are relatively compact in C@,1) with probability1 and the sets of their limit points are S

Proof. This result is a trivial consequence of Theorems 8.2 and 10.2.

Define the process p(xn) @ < x < l;n = 1,2,...) by p(xn) = p* if x =

k/n (k = 0,1,2,..., n) and linear between k/n and (k + l)/n. Then taking into

account that pn is the inverse of f@, n), i.e. f@, pn) = n, we obtain the followingconsequence of Theorem 11.16:

STRONG THEOREMS OF THE LOCAL TIME L21

THEOREM 11.17 The set of limit points of the functions

{2rT2(log log n)p{xn); 0 < x < l} (n -> oo)

consists of those and only those functions f(x) for which f~1(x) 6 Sm .

It is also interesting to characterize the sets of limit points of the sequences

?(x,n) resp. r}(x,t) when we consider them as functions of n resp. t and we

choose a big but not too big x. In fact the Other LIL (cf. Section 5.3) tells us

that

f(zn,n)=0 resp. r](xt,t) = 0 i.o. a.s.

if

resp.\og\ogt

Hence we consider the case when x is smaller than the above limits, i.e. when

f (¦, ¦) and 77(-, ¦) are strictly positive a.s. Now we formulate

THEOREM OF DONSKER AND VARADHAN A977). In the topologyof C(—oo,+oo) the set of limit points of the functions

(/\ 1/2 \

x( ] t\ (t_*oo)\loglogty /

resp.

-—: ) ,n (n -*¦ oo)\ log log nj J J

consists of those and only those subprobability density functions f(x) for which

-r dx < 1.

Remark 1. Mueller A983) gave a common generalization of the Theorem of

Donsker - Varadhan and that of Wichura (cf. Section 8.4).

122 CHAPTER 11

11.5 Stability

Intuitively it is clear that f (z, n) is close to f (y, n) if x is close to y. This Section

is devoted to studying this problem.

THEOREM 11.18 (Csorgo - Revesz, 1985/A).

, N) - ?@, N)_

\Z(k,N)-Z@,N)\

, n) - ?@,n)\

= 2BA: -

where k = ±1, ±2,

THEOREM 11.19 (Csorgo - Revesz, 1985/A).

a.5.

/128N1/4= (ifJ °-

Remark 1. Since for any x € Z1,

?(x, n) = f@, n) i.o. a.s.

the study of the liminf of | ?(x,n) — f@, n) | is not interesting. The limsupproperties of f(z, n) - f@, n) follow trivially from Theorems 11.7 and 11.8.

Theorem 11.18 stated that ?(x, n) is close to f@, n) for any fixed x if n is bigenough. The next two Theorems claim that in a weaker sense ?(x,n) is nearlyequal to f@, n) in a long interval around 0.

THEOREM 11.20 (Csaki - Foldes, 1987). Put

9{t)~

Then

and

lim sup ^-1

limsup sup ^-1

= 0 a.5. if p>2

> 1 a.5. if p<\.

A1.5)

THEOREM 11.21 (Csaki - Foldes, 1987). Put

M+(n)log n (log log n)"' log n(log log n)"'

Then

and

lim supfW00

= 0 a.s. t/ p > -

It

lim sup suprwoo ehi(n)<x<

where c is any positive constant.

-1 = oo a.3. = 0

Remark 2. The Theorem of Hirsch says that ^(x, n) = 0 i.o. a.s. if x >

n1/2(logn)~1. Hence it is clear that A1.5) can be true only if g(n) > n1/2(logn)~1.Theorem 11.20 tells us that g(n) must be smaller than this trivial upper estimate.

Theorem 13.18 will describe the behaviour of ?(M+(n) — j,n) when j is small.

It implies that ?(M+(n) — j,n) is much, much smaller than f@, n). Theorem

11.21 gives the longest interval, depending on Af+(n) and M~(n), where ?(x, n)is stable.

In order to prove Theorem 11.18 we present a few lemmas.

LEMMA 11.1 Let

i = ai{k) = ?{k, Pi) - ?{k, pi-d - 1 (i = 1,2,..., k = 1,2,...).

Then

and

Eax = 0, Ea2 = 4k - 2,

Jim P {n�/2(a1(A:) + at[k) + ¦¦¦ + an{k)) < x{4k - 2I/2}= B7T)-1/2 r e~u2/2du, -oo<x<oo,

J—oo

lim P In-1'2 sup(a1(A:) + a2{k) +•¦¦ + ay (A:)) < x{4k - 2I/21n^°° I J<n J

= {*)V* [*c-»V*du, x>0,7T JO

lim S&

A1.6)

A1.7)

A1.8)

(u.9,

124 CHAPTER 11

Proof. A1.6) is a trivial consequence of Theorem 9.7. A1.7), A1.8) and A1.9)follow from Theorems 2.9, 2.12 and the LIL of Khinchine of Section 4.4 respec-

respectively.The following two lemmas are simple consequences of A1.9).

LEMMA 11.2 Let {//„} be any sequence of positive integer valued r.v.'s with

limn_oo //„ = oo a.s. Then

«!(*)+«,(*) + .. +«,.(*)_ 1/2

rwoo (fin log log fiI'2

LEMMA 11.3 Let {vn} be a sequence of positive integer valued r.v.'s with the

following properties:

(i) lim^oo vn = oo a.s.

(ii) there exists a set flo C fl such that P(ft0) — 0 and for eac^ w ^ fl0 and

k = 1,2,... there exists an n = n(w, k) for which vn{u>,k) = k.

Then

limsup°lW t

n-oo (^

Utilizing Lemma 11.3 with un = ^@, n) and the trivial inequality ot\[k) +a.z[k) +

... + ac@,n)(A:) < ^(/c,n) - ?@, n) < a^/c) + a2[k) +... + aC(o,n)+i(A:) + 1, we

obtain Theorem 11.18.

As far as the proof of Theorem 11.19 is concerned we only present a proof of

the statement

,JV)-{(O,AT) rt28\'"

The other statements of Theorem 11.19 are proved along similar lines.

The proof of Theorem 11.19 is based on the following result of Dobrushin

A955).

THEOREM 11.22

Remark 3. One can also prove that (cf. Theorem 12.1 and A2.17))


This fact together with Theorem 11.22 implies that the rate in the uniform

Invariance Principle (Theorem 10.1) cannot be true with rate n1/4.Dobrushin also notes that if N\ and N2 are independent normal @,1) r.v.'s

then the density function g of \N\\ll2Ni is

2 f°° ( v2 z4\ .

Hence Theorem 11.22 can be reformulated by saying that

(n -> oo). A1.10)

In fact this statement is not very surprising since on replacing n by f@, n)and k by 1 in A1.7), intuitively it is clear that

0~) ,. „

^^

To find an exact proof of A1.11) is not simple at all. We will study this

question in Chapter 12.

Also, by Theorem 9.12

«�/4(e(O,n)I/2 4 W1'2 (n -*. oo). A1.12)

Intuitively it is again clear (for an exact formulation see Chapter 12) that

are asymptotically independent. A1.13)

Hence A1.11), A1.12) and A1.13) together imply A1.10). The proof of Dob-

Dobrushin is not based on this idea. Following his method, however, a slightlystronger version of his Theorem 11.22 can be obtained.

THEOREM 11.23 Let {xn} be any sequence of positive numbers such that

xn = o(logn). Then

^ '

7C J—oo JO

and

P


126 CHAPTER 11

The following lemma describes some properties of the density function g(y). Its

proof requires only standard analytic methods, the details will be omitted.

LEMMA 11.4

(i) There exists a positive constant C such that for any y ? R1

A1.14)

(it) For any e > 0 there exists a C = C[e) > 0 such that

(iii) Let {an} be a sequence of positive numbers with an j oo. Then for any

e > 0 there exist a C\ = Ci(e) > 0 and a C2 = C2{e) > 0 such that

By Theorem 11.23 and (iii) of Lemma 11.4 we have

LEMMA 11.5 For any e > 0 there exist aCx = Ci{e) > 0 and a C2 = C2(e) >

0 such that

P |n-^(e(l,0)- ?(<),»)) > A + 2e) [^)

'

(loglognK/4| < C,(logn)-<1+'>

and

A28\ *f*

^))>(l-2e)( —) (loglogn

Now we prove

LEMMA 11.6

i) . /128N1/4

Proof. Let

nk = (exp(fclogfc)],

f(n) = ?(l,n)-?@,n),, (m, n)) = ?{x, n) - ?{x, m) [m < n),

A* = U{nk)>{l-2e)dk},fa = {l-2e)dk.

By Lemma 11.5

P{At} > C{k\ogk)-^-'\ A1.15)Let y < fc and consider

oo

— A — V* V* P/ At. <-(n A — I 9 — t\

= E EPM* UK) = '.5-

00

00

< E p{?(«*) > /?* - np{f(

t | On. — X

nj) = l}

M = 1, snj =.

.P{?(n,-) = /, Sn

/ / P { f(ij) =

X

,= *}

>Pk- 21/2nJ1/4y}P{f(ni) = 21/2nJ1

/

where

A = %2-WnJ1'* = A - 2*J�/' (^) (log log

and

128 CHAPTER 11

(log log**K'" -

y (=*Now a simple but tedious calculation shows that for any e > 0 there exists a josuch that if j0 < j < k, then

?{AjAk} < (l + e)F{Aj}F{Ak}. A1.16)

Here we omit the details of the proof of this fact, and sketch only the main

idea behind it. Since (nj/n*I/4 < Ar1/4^" = 1,2,...,A; - l), the lower limit of

integration B(y) above is nearly equal to

(^) (loglogn*K/4 if y<k'l\ say.

Hence for latter y values the integral fj?y) g(z)dz is nearly equal to T*{Aic}. Sim-

Similarly, the integral /JJ° g{y)dy gives ~P{Aj}, and A1.16) follows, for in the case of

y > A;1/4 the value of g(y) is very small.

Now A1.15), A1.16) and the Borel - Cantelli lemma combined give Lemma

11.6.

We have also

LEMMA 11.7 Let

mk = [exp(A;/log2A;)]and

Bk = {?@,G71*, m*^)) > ak+1}where

{mk+1 - mk) (log — + 2 log log 771*+!) .

V rnk+i-

m* J\-

m*

Then of the events Bk only finitely many occur with probability 1.

Proof. This lemma is an immediate consequence of Theorem 11.13.

LEMMA 11.8 Let

M*+i = (B + eO7i*+i loglogm*+iI/2and

Dk = I sup sup |a/(l) + a/+1(l) H + a1+y(l)|{l<Mk+i-ak+1 j<ak+1

> V2 [B + e)ak+1 (log^ + \og\ogMk+1^ |Then of the events Dk only finitely many occur with probability 1.

Proof. Cf. Theorem 7.13.

A simple consequence of Lemmas 11.7, 11.8 and Theorem 11.23 is

LEMMA 11.9 Let

Ek = l sup ^(m^n)! > 2B + er)ajfc+i flog—^+loglogMJfc+ij| >.

Then of the events Ek only finitely many occur with probability 1.

LEMMA 11.10

limsuP -T7771—;—hn -̂

V27/

Proof. Let

\ LI /

Then by Lemma 11.5 only finitely many of the events Fk occur with probability1. Now observing that

Mk+l M1/21 I}

log—^1+loglogMJfc+1)| = o{ck),ak+i J\

we have A1.17) by Lemma 11.9 and Lemma 11.10 is proved.Also Lemmas 11.6 and 11.10 combined give Theorem 11.19.

11.6 Favourite points

The random set Jn = {x : ?{x,n) = f(n)} will be called the set of favourite

points of the random walk {S(n)} at time n. The largest favourite points will

be denoted by fn = max{i: x G /„}.Of the properties of {/„} it is trivial that fn < u(n) with probability 1 except

for finitely many n if u(n) G UUCEn). Hence we have a trivial result sayingthat fn cannot be very large. The next theorem claims that /„ occasionally will

be large.

THEOREM 11.24 (Erdos - Revesz, 1984). For any e > 0

with probability 1 infinitely often.

130 CHAPTER 11

Having this result, one can conjecture that /„ will be larger than any function

l[n) i.o. with probability 1 if l(n) € ULCEn).However, it is not the case. Conversely, we have

THEOREM 11.25 (Erdos - Revesz, 1984).

fn < (nB log2 n + 3 log3 n + 2 log4 n + 2 log5 n + 2 log6 n

with probability 1 except for finitely many n.

It looks also interesting to investigate the small favourite points. Let gn =

min{|x| : x 6 /„}. Bass and Griffin A985) proved that gn cannot be very

small. In fact

THEOREM 11.26

joo if 1>U,

)--* \0 if -y<2.

Here we present a few unsolved problems (Erdos - Revesz, 1984 and 1987).

1. Theorem 11.24 stated that /„ > (I — e)b~1 infinitely often with probability1. Its proof shows that when /„ > (l — e)b~1, then f (/„, n) = f (n) will be

larger than Db~l (where D is a small enough positive constant) infinitelyoften with probability 1. It is not clear how big f(n) can be when /„ >

A — e)b~1 or how big /„ can be when f (n) > A — ^b'1.

2. Everyone can see immediately that \7n\ = 1 and \7n\ = 2 i.o. with proba-probability 1. Can we say that |/n| > 3 infinitely often with probability 1?

3. Consider the random sequence {un} for which |/i,n| > 2. What can we say

about the sequence {^n}? Can we say, for example, that limn—oo vn/n = oo

with probability 1?

4. How can the properties of the sequence |/n+i — fn\ be characterized? Is it

true that limsup,^^ |/n+i — fn\ = oo? If yes, what is the rate of conver-

convergence?

5. Does the sequence /n/\/n have a limit distribution? If yes, what is it?

6. Let ot(n) be the number of different favourite values up to n, i.e. a(n) =

| Z))b=i ?k\- We guess that ct(n) is very small, i.e. a(n) < (logn)c for some

c > 0, but we cannot prove it. Hence we ask: how can one describe the

limit behaviour of a(n)?

7. We also ask how long a point can stay as a favourite value, i.e. let 1 < i =

*(n) < j — ]{n) < n be two integers for which

and j — i = /?(n) is as big as possible. The question is to describe the limit

behaviour of /?(n).

8. Further if x was a favourite value once, can it happen that the favourite

value moves away from x but later returns to x again, i.e. do sequences

an < bn < cn of positive random integers exist such that

?„/>„= 0 and /an/Cn^0 (n = l,2,...)?

9. To investigate the jumps of the favourite values looks also interesting. Let

n = n(w) be a positive integer for which 7n?n+i — 0- Then the jump-sizejn is defined as

jn = p(?., ?.+i) = min{|x-y|; x € /n,y € /n+i}.

The theorem of Bass and Griffin (Theorem 11.26) implies that jn >

n1/2(logn)~11 i.o. a.s. It looks very likely that limn—oojn = oo a.s. We do

not see how one can describe the limit behaviour of jn.

10. By the arcsine law we learned that the particle spends a long time on one

half of the line and only a short time on the other half with a big probability.We ask whether the favourite value is located on the same side where the

particle has spent the long time. For example let 0 < ti\ < ni(w) < n-i...

be a random sequence of integers for which

where

J 1 if S' > 0

Then we conjecture that /nt —*• oo as A; —*• oo a.s.

132 CHAPTER 11

11.7 Rarely visited points

It is easy to see that for infinitely many n almost all paths assume every value

at least twice which they assume at all, i.e. let 6^ = 0 if f@, n) ^ r - 1 and

<$ir) = 1 if f@,n) = r - 1 and let

be the number of points visited exactly r-times up to n. Then

P{/1(n)=0i.o.} = l.

We do not know if for infinitely many n almost all paths assume every value at

least r-times (r = 2,3,...) which they assume at all, i.e. let

and we ask

P{(/r(n)=0i.o.} = ?

We would guess that this probability is 0 if r > 2 but perhaps it is 1 if r = 2.

A study of

liminf/r(n) and limsup/r(n) (r = l,2,...)n-K» n-K»

looks also interesting.As already stated liminfn—oo /i(n) = 0. Major A988) proved

THEOREM 11.27

limsupl

,= C a.s.

n—oo log n

where 0 < C < oo but its exact value is unknown.

Another interesting result on /i(n) is

THEOREM 11.28 (Newman, 1984).

E/!(n)=2 (n = l,2,...).Proof. Since /x(l) = 2 we only prove that E/^n + l) = E/Jn) for n > 0.

Consider the walk S? = Sk+i — Xi (k = 0,1,2,...) and let '/i (n) ^e t^ie number

of points visited exactly once by S? up to n. Then

/*(n) + l if ?@,n+l)=0,/i(n)-l if ?@,n+l) = l,

K{n) if e@,n+l)>l.Theorem 9.3 implies that P{^@,n + l) = 0} = P{?@,n + 1) = 1}. Hence we

have the Theorem.

Chapter 12

An embedding theorem

12.1 On the Wiener sheet

Let {Xij, i = 1,2,...,y = 1,2,...} be a double array of i.i.d.r.v.'s with

and define S0>n = Sm<0 = 0 (n = 0,1,2,...; m = 0,1,2,...},

7=1i=l

The arrays {S^m} and {Xij} are called random fields. Some properties of {S^m}can be obtained as simple consequences of the corresponding properties of the

random walk, some properties of {5>,,m} are essentially different. Here we men-

mention one example of both types. Just like in the one-dimensional case we have

lim p(^^ <*} = $(*).{/ Jn-~ {y/nrn

However,

Iimsup6(nmMnm = 21/2 a.s.

m—t-oo

This latter result is due to Zimmermann A972) (see also Csorgo - Revesz, 1981).On the same way as the Wiener process was defined (Section 6.2) a continuous

analogue of {Snm, n = 0,1,2,...; m = 0,1,2,...} can be defined. This contin-

continuous random field will be called Wiener sheet (two-parameter Wiener process)and will be denoted by

{W{x,y), x>0,y>0}.

Among the properties of the Wiener sheet we mention

133

134 CHAPTER 12

(i) W(-, •) is a Gaussian process,

(ii) W{0,y)=W{x,0)=0,

(Hi) EW(xuy1)W(x2, y2) = min(xi,x2) min(yi,y2),

(iv) VT(x,y) is continuous a.s.,

— 1/2

(v) for any x0 > 0, the one-dimensional process {xQ W{x0, y), y > 0} is a

Wiener process,

(vi) for any y0 > 0, the one-dimensional process {y0 W(x,y0), x > 0} is a

Wiener process.

For some further study and a detailed definition of the Wiener sheet we refer

to Csorgo - Revesz A981).

12.2 The theorem

We have already seen that the study of the processes ?(x,n) resp. r}(x,t) is

relatively easy when x is fixed and we let only n resp. t vary. The main reason

of this fact is the following trivial:

LEMMA 12.1 For any integer x

Z{x,Pi) - ?{x,p0) = Z[x,pi),Z{x,p2) - ?{x,pi),Z{x,p3) - ?(x,p2),...

are i.i.d.r.v.'s with

E(e(*,P*) - e(*,p*-i)) = l,E(?(z,pfc) - ?(*,p*-i) - IJ = 4x - 2

(cf. Theorem 9.7).

In order to formulate the analogue of Lemma 12.1 for rj(-, ¦) let

Po=O, p: = mf{t;t>0,V{0,t)>u} (u > 0). A2.1)

Then we have

LEMMA 12.2 For any x ? R1, r)(x,p*u) is a process of independent increments

in u(u > 0), i.e. for any 0 < ui < u2 < ... < ujt(A; = 1,2,...), the r.v. 's

AN EMBEDDING THEOREM 135

are independent with

where j = 1,2,..., A;.

Consider the process

?(x,u) = ^(x.pi) - 7/@,P;) = ^(x,P;) - u. A2.2)

Then we have

(i)E?(x,u)=0, E?2[x,u) =4xu,

(ii) {?(x, u);u > 0} is a strictly stationary process of independent increments

in u for any x 6 J?1.

One can also prove that

(iii) ?(x, u) has a finite moment generating function in a neighbourhood of the

origin.

By the Invariance Principle 2 (cf. Section 6.3) this fact easily implies that for

any iGi?1 the process ?(x,u) can be approximated by a Wiener process W*(-)with rate O(logu), i.e.

Having a fixed x this result gives an important tool to describe the properties of

C{x,u).What can we say about ?(x,u) when u is fixed and x is varying? It is easy

to prove that for any fixed u {?(x,u),x > 0} has orthogonal increments and it

is a martingale in x. This observation suggests the question:Can the process ?(x, u) be approximated by a two-parameter Wiener process?Since by the LIL r}(x, u) = 0 a.s. if x > (B + ejulogloguI/2 and u is big

enough, we have ?(x, u) = —u for any x big enough. This clearly shows that

the structure of ?(x, u) is quite different from that of W(x, u) whenever x is big.Hence we modify the above question as follows:

Can the process ?(x, u) be approximated by a Wiener sheet provided that u

is big but x is not very big?The answer to this question is positive. In fact we have


136 CHAPTER 12

THEOREM 12.1 (Csaki - Csorgo - Foldes - Revesz, 1989). There exists a

probability space with

(i) a standard Wiener process {W(t),t > 0}, its two-parameter local time pro-

process {rj(x, t),x 6 Rl,t > 0} and the inverse process p*u of 77@,*) defined by

A2.1),

(ii) a two-time parameter Wiener process {W(x,u);x > 0, u > 0} such that

sup \?{x,u) -2W[x,u)\ =o(ui?-e) a.s. [u -* 00) A2.3)0<x<Aus

where ?(x, u) is defined by A2.2), A is an arbitrary positive constant and

0 < 6 < 7/100,0 < e < 1/72 - 6/7.

This theorem is certainly a useful tool for studying the properties of ?(x, u) or

r}(x,p*u). Unfortunately it does not say too much about rj(x,t). However, we can

continue Theorem 12.1 as follows:

THEOREM 12.2 On the probability space of Theorem 12.1 we can also definea process p*u such that

{p>>0}?{p;,u>0}, A2.4)

\PI - P*u\ = O (u15/8) a.s. (u-00), A2.5)

{Pu>« > 0} and {W(x,u);x>0,u> 0} are independent. A2.6)

Having the process {p*, u > 0} we can proceed as follows:

Define the local time process rj{O,t) by

By the continuity properties of 77@,*) (cf. Theorem 11.7) we have

= 0 a.s.

Thus by Theorem 12.2 we conclude that the local time process {77@,*);* > 0}has the following properties:

{rj{O,t);t > 0}Mv{0,t);t > 0}, A2.7)

1*7@,*) - v[O,t)\ is small a.s. (t -* bo), A2.8)

{17@,t);t >0} and {W(x,u);x >0,u > 0} are independent. A2.9)

A2.7) resp. A2.9) follows immediately from A2.4) resp. A2.6). In order to see

A2.8) it is enough to show that

is small, which in turn follows from the fact that

is small. Now A2.3), A2.8), and the continuity of W(-, •) imply

\rj{x,t) -77@,*) -2W(x,f)@,t))\ is small a.s.

where 77@,*) satisfies A2.8) and A2.9).A precise version of the above sketched idea implies

THEOREM 12.3 (Csaki - Csorgo - Foldes - Revesz, 1989). There exists a

probability space with

(i) a standard Wiener process {W(t);t > 0} and its two-parameter local time

process {r)(x,t);x E R},t > 0},

(ii) a two-time parameter Wiener process {W(x,u);x > 0,u > 0},

(iii) a process {f){O,t);t >0}={r}{0,t);t > 0}

such that

sup \n{x,t)-r,{0,t)-2W{x,f,{0,t))\=o(ti?->) a.s. (t -» 00),<AT*/2

a.s. (t -> 00),

{17@, t); t > 0} and {W(x, u); x > 0, u > 0} are independent

where

A > 0,0 < 6 < 7/100, 0 < e < 1/72 - 6/7.

12.3 ApplicationsIn order to show how the above theorem can be used in the study of the propertiesof 77(-, •), first we list a few simple properties of the vector valued process

138 CHAPTER 12

which can be obtained by standard methods of proof.Namely for any x > 0 and t > 0 we have

where Ni,N2 are independent normal @,1) r.v.'s.

Also, for any x > 0, the set of limit points of

Ut=

is the interval [—1,1] a.s. The set of limit points of

y

t-l'*f)(O,t) I \N2\, A2.11)

llNl\N^ A2.12)

y/2t log log t

is the interval [0,1] a.s. The set of limit points of

(Ut,Vt)

is the semidisc {{u,v) : v > 0,u2 + v2 < 1}. The set of limit points of

rri/i/2 w(M(o,0)

is the interval [0,21/23�/4] a.s. for any x > 0, that is,

The usual LIL implies

i- W(x,fj@,t))hmsup sup . ===== = 1 a.s.

«-»«, o<z<Kt* yj2Kri @, t)ts log log t

Applying again the independence of Ut and Vt we obtain

for any K > 0 and 6 > 0.

Consequently, by Theorem 12.3 and by A2.10), A2.11), A2.12), A2.13),A2.14) respectively we obtain

2yJxr,{0,t)0

limsup i/iw!n?@>/?3/4 = ^6l/4 a-s-

x1/2i1/4(lltK/4 3

i- r,{x,t) - r,{O,t)limsup sup

*(x,t) - r,(O,t) ± ^ forany x>Q ^^

a| (t >0), A2.16)

^\1'2 *->o0' forany

limsup—yifL=L===2LjJ== = 1 a.s. forany x > 0, A2.18)*-«» 2yJ2xr)@, t) log log t

sup ,=

0<x<Kt* 2^/2Ktsri@, t) log log t

i-3 1/4 n(x,t) - r}(O,t)

limsup sup -6-1'* . ,' /.n> j

/4=1 a.s. 12.20

for any /f > 0 and 0 < 6 < 7/200.For the direct proofs of A2.18) and A2.19) see Csaki - Foldes A988).

Chapter 13

Excursions

13.1 On the distribution of the zeros of a

random walk

(9.11) and Theorem 11.1 are telling us in different forms that f@, n) convergesto oo like n1/2, i.e. the particle during its first n steps visits the origin practicallyn1/2 times. Clearly these n1/2 visits are distributed in [0,n] in a very nonuniform

way. We have already met the Chung - Erdos theorem (Theorem 11.10) and

the arcsine law (9.12) claiming that the zeros of {Sk} are very nonuniformlydistributed at least for some n. Now we give a few reformulations of the Chung- Erdos theorem in order to see how it describes the nonuniformness of the

distribution of the zeros of {Sk}. First a few notations:

(i) let

R{n) = max{A; : k > 1 for which there exists a 0 < j < n - k

such that f@,y + A;) - f@, j) = 0}

be the length of the longest zero-free interval,

(ii) let

R{n) = max{A;: k > 1 for which there exists a 0 < j < n - A;

such that M+{j + k)= M+{j)}

be the length of the longest flat interval of M? up to n,

(iii) let

#(n) = max{Jt : 1 < k < n, Sk = 0}be the location of the last zero up to n,

141

142 CHAPTER 13

(iv) let <;n be the number of those terms of 5j, 52,..., Sn which are positive or

which are equal to 0 but the preceding term of which is positive,

(v) let

fi+(n) = inf{Jt : 0 < k < n for which Sk = M+}.

Now we can reformulate the Chung - Erdos theorem (Theorem 11.10) as

follows:

THEOREM 13.1 Let f(x) be a nondecreasing function for which lim^oo f(x)= oo, x/f(x) is nondecreasing and lim^oo x/f(x) = oo. Then

if and only ifr°° dx

h

where Y(n) is any of the processes R(n),R(n),n — \&(n),?n,n — fi+(n).

Proof. It is immediately clear that

UUC(i?(n)) = UUC(n - *(n))

and

UUC(JR(n)) = UUC(n - fi+{n)).

By Theorem 10.3 it is also clear that

UUC(J2(n)) = UUC(^(n)).

As far as the process $n is concerned we clearly have

Cn) C UUC(n - V{n)).

The equality in the last relationship is not quite clear but following the originalproof of Theorem 11.10 given by Chung and Erdos A952) we get the requiredequality.

The characterization of the lower classes of n — \&(n) is trivial since we have

= n i.o. a.s.

The characterization of the lower classes of $n is also trivial. In fact as a simpleconsequence of Theorem 13.1 we obtain

EXCURSIONS 143

THEOREM 13.2 Assume that f(x) satisfies the conditions of Theorem 13.1.

Then

7^7 € LLC(fn)

i/ and on/y i/

z(/(z))i/'<0°-

The characterization of the lower classes of i2(n) and R(n) is much harder. We

have

THEOREM 13.3 (Csaki - Erdos - Revesz, 1985). Let f(x) be a nondecreasingfunction for which

/(x)/oo,—/oo (z->oo).

Then

if and only if

n=l

where Y* is any of the processes R(n) and R(n) and E = 0,85403... is the root

of the equationoo

yk=l

Consequence 1.

. log log n. log log nblim inf R(n) = lim inf R(n) = 3 a.s.()

Besides studying the length of longest excursion R(n), it looks interesting to say

something about the second, third,... etc. longest excursions. Consider the sam-

sample Pi,p2-

Pi, ••• iP€(o,n) -Pc(o1n)-n»»-Pc(o1n) (the lengths of the excursions) and

the corresponding ordered sample Ri{n) = R(n) > R2{n) > ... > i?f@,n)+i(n).Now we present

THEOREM 13.4 For any fixed k = 1,2,... we have

144 CHAPTER 13

This theorem in some sense answers the question: How small can R2(n),R3(n),...be? In order to obtain a more complete description of these r.v.'s we present the

following:

Problem 1. Characterize the set of those nondecreasing functions /(n) (n =

1,2,...) for which

Theorem 13.1 tells us that for some n nearly the whole random walk {S(k)}%=0is one excursion. Theorem 13.3 tells us that for some n the random walk consists

of at least 0~l log log n excursions. These results suggest the question: For what

values of k = k(n) will the sum Z)*=1-R;-(n) be nearly equal to n? In fact we

formulate two questions:

Question 1. For any 0 < e < 1 let T(e) be the set of those functions f(n) (n =

1,2,...) for which()

with probability 1 except finitely many n. How can we characterize /(e)?

Question 2. Let J{o) be the set of those functions f{n)(n = 1,2,...) for which

lim n1 y -R.-(n) = 1 a.s.

n—»oo*—' ' v '

How can we characterize /(o)?Studying the first question we have

THEOREM 13.5 (Csaki - Erdos - Revesz, 1985). For any 0 < e < 1 there

exists a C = C(e) > 0 such that

Clog logn€ J{e).

Concerning Question 2, we have the following result:

THEOREM 13.6 For any C> 0

/(n) = C log log n$f{o)

and for any h(n) /" oo (n —> oo)

/i(n) loglogn € J(o).

EXCURSIONS 145

Knight A986) was interested in the distribution of the duration of the longestexcursion of a Wiener process. In order to formulate his results introduce the

following notations: for arbitrary i>Owe set

to(t) = sup{s : s < t,W(s) = 0},

ti(t) = inf{s : s > t,W(s) = 0},

d{t) =*!(*) -to{t),D{t) = sxxp{d{s) : to(s) < t},

E(t) = snp{d(s) : s < t,ti{s) < t}.

Then we call d(t) the duration of the excursion containing t. D(t) resp. E(t) is

the maximal duration of excursions starting by t resp. ending by t.

Knight evaluated completely the Laplace transforms of the distributions of

D(t) and E(t) and the distributions themselves over a finite interval. His results

run as follows:

THEOREM 13.7 (Knight, 1986).

where

Ft ^_/27r"^1/2 «/ y^1'

*W-\x-l(Z-y+j.\Ogy) if l<y<2

and

where G{1) = 0,

if i<y<2,if 2<y<3,

and

GB) = I - i.^7T 2

The multiple Laplace transform of D(t) and some other characteristics of a

Wiener process were investigated by Csaki - Foldes A988/A). A very different

characterization of the distribution of the zeros of {Sn} is due to Erdos and

Taylor A960/A), who proved

146 CHAPTER 13

THEOREM 13.8

1n

lim y^Pt = T~ as.»-"» log n

~

Remark 1. (9.8) and Theorem 11.6 claim that pk converges to infinity like k2.

However, these two results are also claiming that the fluctuation of k~2pk can be

and will be very big. Theorem 13.8, via investigating the logarithmic density of

pj[ , also tells us that pk behaves like k2.

Let us mention a result of Levy A948) that is very similar to the above

theorem.

THEOREM 13.9

lim : >,

= - a.s.«-»«» log n ^ k 2

wherefl •/ Sk >0,

-{o ,/ sk<o.

Remark 2. Theorems 13.1 and 13.2 imply that

1n

liminf-V I(Sk) = 0 a.s.

and1

n

lim sup— ^2 I{Sic) = 1 a.s.

noo ftftfc=1

Hence the sequence /(S*) does not have a density in the ordinary sense but byTheorem 13.9 its logarithmic density is 1/2.

It is natural to ask what happens if in Theorem 13.9 the indicator function

/(•) of (—oo,0) is replaced by the indicator function of an arbitrary Borel-set of

R1. We obtain

THEOREM 13.10 (Brosamler, 1988; Fisher, 1987 and Schatte, 1989). There

is a P-null set N C fl such that for allu <? N and for all Borel-sets A C R1 with

X(dA) = 0 we have

lim

EXCURSIONS 147

where dA is the boundary of A and

For a Strassen type generalization of Theorem 13.10, cf. Brosamler A988) and

Lacey - Philipp A989).For the sake of completeness we also mention

THEOREM 13.11 (Weigl, 1989).

J \ (?,k->I(Sk) - ilogn) < xl = *(x)

where

/I fo

/(•) is defined in Theorem 13.9.

13.2 Local time and the number of longexcursions (Mesure du voisinage)

The definition of the local time of a Wiener process (cf. Section 9.3) is extrinsic

in the sense that given the random set At = {t : 0 < t < T,W(t) =0} one

cannot recover the local time rj(O,T). Levy called attention to the necessity of

an intrinsic definition.

He proposed the following: Let N(h,x,t) be the number of excursions of

W(-) away from x that are greater than h in length and are completed by time

t. Then the "mesure du voisinage" of W at time t is \imhs^0 h1?2N(h, x,t), and

the connection between rj and N is given by the following result of P. Levy (cf.Ito and McKean 1965, p. 43).

THEOREM 13.12 For all real x and for all positive t we have

lim h^2N{h, x, t) = \\-ri{x, t) a.s.

h,"\0 y 7T

Perkins A981) proved that Theorem 13.12 holds uniformly in x and t. Cso^goand Revesz A986) proved a stronger version of Perkins' result. Their results can

be summarized in the following four theorems.

148 CHAPTER 13

THEOREM 13.13 For any fixed t' > 0 we have

hl'2N(h,x,t)-J-r,(x,t)V 7T

sup(x,t)eRlx[O,t'}

= 0 a.s.

The connection between N and 77 is also investigated in the case when a Wiener

process through a long time t is observed and the number of long (but much

shorter than t) excursions is considered. We have

THEOREM 13.14 For some 0 < a < 1 let 0 < at < ta(t > 0) be a nonde-

creasing function oft so that at/t is nonincreasing. Then

Tlim — log —

sup<-°° \t J \ aj XfzRi

N{at,x,t)-J r,{x,t) = 0 a.s.

The proofs of Theorems 13.13 and 13.14 are based on two large deviation type

inequalities which are of interest on their own.

THEOREM 13.15 For any K > 0 and t' > 0 there exist a C = C{K,t') > 0

and a D = D{K,t') > 0 such that

sup >C\ < DhK,

where h < t'.

THEOREM 13.16 For any K > 0 there exist aC = C(K) > 0 and a

D = D{K) > 0 such that

log -at

sup N{at,x,t)-J r,{x,t) >C

where 0 < at < t.

It is natural to ask about the analogues of the above theorems for random walk.

Clearly for any x = 0,±1,±2,... the number of excursions away from x

completed by n is equal to the local time f(x, n), i.e.

M(x, n) = {the number of excursions away from x completed by n} =

max{t : pi{x) < n} = f(x, n).Hence we consider the following problem: knowing the number of long excur-

excursions (longer than a = an) away from x completed by n, what can be said

about f(x,n)? Let M(a,x,n) be the number of excursions away from x longerthan a and completed by n. Our main result says that observing the sequence

n,x,n)}^L1 with some an = [na]@ < a < 1/3) the local time sequence

n)}^=l can be relatively well estimated. In fact we have

EXCURSIONS 149

THEOREM 13.17 Let an = \na\ with 0 < a < 1/3. Then

/a\ 1/4 / n\-llim — 1 (log—I sup \M(an,x,n) - ?(x,n)P(an)\ = 0 a.s.n->°° V n / V an) x€Zi

where

P[a) = P{pj > a}.

The proof of this theorem is based on

THEOREM 13.18 For any K > 0 there exist a C = C(K) > 0 and a D =

D(K) > 0 such that

Kn\ 1/4 / _ \ -3/4 1

-^ log— sup \M{an,x,n) - Z{x,n)P{an)\ >C\< Dn~K,n / \ an/ X?zl )

where an = [na] @ < a < 1/3).

Remark 1. Very likely Theorems 13.17 and 13.18 remain true assuming onlythat 0 < a < 1.

In order to prove Theorem 13.18, first we prove the following simple

LEMMA 13.1 Let n and a be positive integers and C > 0. Then

M{a,x,pn{x)) -nP(a)(nP(a)(l-P(a))lognP(a)I/2 >C1/2|<2(P(a)n)-2C/9,

provided that

Clog(nP(a)) A3.1)

Proof. Clearly M(a, x,pn(x)) is binomially distributed with parameters n and

P(a). Hence the Bernstein inequality (Theorem 2.3 ) easily implies Lemma 13.1.

Proof of Theorem 13.18. Since by (9.10)

.-i'

condition A3.1) holds true if a < np(p < 2) and n is big enough, Lemma 13.1

can be reformulated as follows: for any K > 0 and 0 < if) 0 and D = D{^,p,K) > 0 such that

M{a,x,pn{x)) - Z{x,pn{x))P{a)2 \

—)7ra/

>C Dn~KA3.2)

150 CHAPTER 13

provided that n* < a < np.

A3.2) in turn implies

supM{a,x,pn{x)) -

>C \ < Dn~K, A3.3)

and for any K > 0, 0 < V = D(t,6,ip,p,K) such that

sup supM{a,x,pn{x)) - Z{x,pn{x))P{a)

„!/*(-?-)'

(logna

y>C -K. A3.4)

Then by a slight generalization of (9.11) (or applying the exact distribution of

f@,n), cf. Theorem 9.3) for any K > 0 there exist a C = C(K) > 0 and a

D = D{K) > 0 such that

Clogn<Dn~K A3.5)

for any x € Zl or equivalently

P{f(z,n) >C(nlognI/2} < fln"*. A3.6)

Let m be a fixed positive integer and assume that the event

Am = [m? < ?{x,m) < C(mlogmI/2} @ < /? < 1/2)

holds true. Then replacing m by pn(x) (more exactly assuming that f (x,m) = n,

i.e. pn{x) < m < pn+i(x)) we obtain

J = J(m,x) =M(am,x,m) - ?{x,m)P{am) M{am,x,pn{xj) - nP(am)

where by the assumption that Am holds true we have

m? < ^(x,m) = n < C(m log mI/2,

i.e.

2C2 log n

m

EXCURSIONS

Hence

151

J < supmP<n<C(m log mI/2

M(am,x,pn(x)) -nP(am)1/4

2C2 log n

-1/4log

n3/4

am2C2logn

< 4 sup supM(a,x,pn(x)) -nP(a)

Observe that if f (x, m) < m^ then

J <mh

Consequently

0 if 0<

=0

aJ

1-a

A3.7)

A3.8)if m is big enough and /? < A - a)/4. Hence by A3.8), A3.7) and A3.4) we

obtain

P{J >C} = P{J > C, Am} + P{J > C, f (x, m) < mp)+ P{J > C,^(x,m) > C(mlogmI/2} < P{J > C, Am}+ P{^(x,m) >C(mlogmI/2}

< P{J > C, Am} + Dm~K < 2Dm~K

if m is big enough, /? < ^-^ and | < 2. /? can be chosen in such a way if

0 < a < 1/3. Consequently we have also that

-KP{sup J(m,x) > C} < Dm

for any K > 0 if C, D are big enough and 0 < a < 1/3. Hence the proof of

Theorem 13.18 is complete.Theorem 13.17 is a trivial consequence of Theorem 13.18.

Note that if a > 1/5 then P(a) can be replaced by B/7raI/2. Hence we also

obtain

THEOREM 13.19 Let an = [na] with 1/5 < a < 1/3. Then

nsupz€Zl

M(onii,n)-((i,n) )\nan/

J/2= 0 a.s.

152 CHAPTER 13

THEOREM 13.20 For any K > 0 there exist a C = C{K) > 0 and a D =

D(K) > 0 such that

( /a \ !/4 / n \ ~3/4 / 2 \P (-) (log-) sup M(an,s,n)-f(s,n)( )

w/icrc an = \na\ A/5 < a < 1/3).

> C < Dn~KJ

13.3 The local time of high excursions

Theorem 9.7 described the distribution of the local time f (A:,pi) of the excursion

{S0,Si,... ,SPl}. Now we are interested in the properties of f(A:,pi) when k is

big, i.e. when k is close to M+(pi), the height of the excursion {So, Si,..., SPl}.We are especially interested in the limit distribution of f(A:,pi) when k is close

to M+(pi) = n and n —> oo. First we present the simple

THEOREM 13.21 For any n = 1,2,... and / = 1,2,... we have

P{M+(Pl) = n, f(n,Pl) = /} = n-22-'-

if n ^ oo then

= n} = 5±i (^i) - 2"',

Proof. By Lemma 3.1 the probability that the excursion {So, Sx,..., SPl} hits n

is Bft)-1, i.e. P{M+(pj) > re} = Bft)-1. The probability that after the arrival

time Pi(n) the particle turns back but hits n once more before arriving at 0 is

1/2A — 1/n). Hence the probability of / — 1 negative excursions away from n

before px is A/2A - 1/n)I�. Finally Bft)-1 is the probability that after / - 1

excursions the particle returns to 0.

In order to study the properties of ?(M+(p\) — j,pi), first we investigate the

distribution of f(M+(pj) - j,Pi{M+(pi))). (Note that p1(M+(p1)) is the first

hitting of the level M+(p\).)

EXCURSIONS 153

LEMMA 13.2 For any I = 1,2,..., n = 1,2,..., j = 1,2,... ,n- 1

(Ww h

-i) V 2j{n-j)

and if n —> oo t/icn

n ( n

1

2j(n-j)\ 2j{n-j)J 2j \ 2jJ

> n) = > 2j,n

— \M+{px)>n

n n

Proof.

z n —

Further,

is the probability that after p\{n — j) the particle makes u negative excursions

away from n—j (none of them reaches 0) and / — 1 — u positive excursions away

from n—j (none of them reaches n) in a given order. Finally By)-1 is the

probability that after the / — 1 excursions the particle goes to n.

LEMMA 13.3 For any n = 2,3,..., / = 2,3,..., j = 1,2,..., n - 1

(« -i.Pi) = I I A/+(p,) = n,e(n,pi) = 1} = P{?/, + U2 = 1}

4?


154 CHAPTER 13

where U\ and Ui are i.i.d.r.v.'s with

m-1

Further,

and

n

,...). A3.9)

(n^oo)

•n\ 2

n

n n

Proof. Since

and by Lemma 13.2 the conditional distribution of f(n — j,p\(n)) and that of

^(« - J,Pi) ~ Z{n-J,Pi{n)) (given (M+(pi) = n,^(n,p!) = 1}) are equal to the

distribution of U\ we obtain Lemma 13.3 realizing that the two terms of righthand side of A3.10) are conditionally independent.

LEMMA 13.4 For any n = 2,3,... ,j = 1,2,... ,n- 1 and I = 0,1,2,...

l-y-1

i1-

if 1 = 0,i-i

n+1 (

n

if n —^ oo then

t/ /=0,

n+1

n(n.(n + l-jJ^2,


EXCURSIONS 155

E

n + 1 - j

n(n 2(n + 1 - j) n8j - 6.

Proof is essentially the same as that of Lemma 13.2.

LEMMA 13.5 For any n = 2,3,..., j = 1,2,..., n - 1, k = 1,2,... and

1 = 2,3,...

-.7,Pi) = ' I M+{Pl) = n.tfn.pi) = A:}

+ vl + v2 + ..- + vk_l + u2 =

where U\, Vy, V2,..., Vk-\,U2 are independent r.v.'s with

/ \ rn-\

i = m} = ^727^rT f 1 7r2.7A1-j) V 2j(n-j)

F{V{ =m} =

j) n-1

t = l,2;m = l,2,...)

if m = 0,

n

1- rr

nm-l

if m > 1,

n-1 n(n-l)

- 1 -j) - n

n-1 n-1

- Aj - 6) = 8j2 + {2k - 3Lj - 6(ik - 1).


156 CHAPTER 13

Proof. Clearly

fc-i

t{n-J,Pi) = t{n-J,pM) + E(f(n -i.P<+i(n)) - t{n-J,Piin))t=i

+ {t{n-j,Pi) ~ Z{n-j,pk{n))

where

?(« " i.PiH), e(« " i,P.-+i(»)) " ?(» " i,P.'(»)) (i = 1,2, ...,*- 1)

and

^(n-JiPi) -^(n-JiPfcH)

are independent. Lemma 13.3 tells us that the conditional distributions of f (n —

j,p\(n)) and f (n — j,pi) — ?(n —j,pk{n)) are equal to the distribution of U\ and

U2. Lemma 13.4 claims that the conditional distributions of f(n - j,pt+1(n)) -

?{n—j\pi(n)) (i = 1,2,..., k — 1) are equal to the distribution of Vj, V2,..., V^j.Hence we have Lemma 13.5.

Theorem 13.21 and Lemmas 13.3, 13.4 and 13.5 combined imply

THEOREM 13.22 For any j = 1,2,..., n - 1; n = 2,3,...

(„ + 1 - j)^ 4, + 2,LZ& + (^ l)n Vn + 1 / n(n + 1)

n - j,Pl) - Ef(n - y,Pl)J | M+(Pl) = n) - 8j2 + 4j - 6.

Further, for any j = 0,1,2,... and K > 0 there exist a C\ = C\(K,j) > 0 and a

C2 = C2{K,j) > 0 such that

P {?(" -J,Pi) > Ci log" | M+(Pl) = n} < Cjn"*

/or any a > 0 and if > 0 there exist a C\ = Ci(a,K) > 0 and a C2 —

C2{a,K) >0 such that

P{^(n-alogn,p!) >C1log2n|M+(p1) = n} < Czn'^.

We also obtain the following:

Consequence 1. For any j = 0,1,2,..., n and n big enough we have

P {f (n " i.Pi) > 6J2 + 4j + 2 | M+(Pl) =n}<±.

EXCURSIONS 157

Proof. By Chebyshev inequality and Theorem 13.22 we have

P {tfn -i.Pi) > A(8j2 + Aj - 6I/2 + Aj + 2\ M+(Pl) = n} < 1.

Taking A = 2j and observing that

2j (8j2 + Aj - 6I/2 + Aj + 2 < 6j2 + Aj + 2

we obtain the above inequality.

13.4 How many times can a random walk

reach its maximum?

Let x{n) be the number of those places where the maximum of the random walk

SO,SU... ,Sn is reached, i.e. x(n) is tne largest positive integer for which there

exists a sequence of integers 0 < k\ < k2 < ¦.. < kx(n) < n such that

S(*i) = S(k2) = ¦¦• = S(kx{n)) = M+(n). A3.11)

Csaki (personal communication) evaluated the exact distribution of x(n)- In

he obtained

THEOREM 13.23 For any k = 0,1,2,..., [n/2]; n = 1,2,... we have

P{X(n) = k + 1} = 2-*P{AC_t > *}.

Proof. Consider the sequence

2),X(kx[n)-l + 3),..., X(kx{n))},

where X(l) = Xt = S(l + 1) - 5(/) and A:,, Jt2,... ,kx{n) are defined by A3.11).Let Sj (j = 0,1,2,... ,n

— x(n) + 1) be the sum of the first j of the above

given random variables in the given order. Then {Sy} is a random walk and

x(n) = k + 1 if and only if maxo<j<n-jfc Sj > k which implies the Theorem.

Now we prove a strong law.

158 CHAPTER 13

THEOREM 13.24

maxi<jt<nx(fc) 1hm r-= = -

a.s.,n->oo Jg n Z

consequentlyX(n) 1

limsup 7-^—- = - a.s.

n-oo lgn 2

and triviallyx(n) = 1 i.o. a.s.

Proof. Consider the sequence

Then

max y(A;) = max ?.k<n i<M+

ClearlyP{? = k}=2~k {k = 1,2,...; t = 0,1,...)

and the random variables f* (t = 0,1,...) are independent. Hence for any

L = 1,2,... and K = 1,2,... we have

Choosing

K = Kn = ——— lg n and L = Ln =

It

we obtain

PI t^n

if n is big enough. Hence

max ft* 1 _

Hminf — > —-— a.s.n—oo lg n 2

Since M+ > n1/2(lgn)� a.s. (cf. Theorem of Hirsch, Section 5.3) for all but

finitely many n we get

hminf r= — > - a.s.n—oo lgn 2

EXCURSIONS 159

Similarly, choosing

K = Kn=]-^lgn and L = Ln = n1/2\gn

we get/ 1 lgn

p{ig«e; > if} = 1-A-^57) «

Let n^T) = >T. Then we have

max ?,* < K a.s.i<L

l ~~

for all but finitely many j where K = Kn.(T) and L = Ln^T). Let jT < N <

{j + 1)T. Then1 +2e

max ?t* < Kn-+1iT) ^ —« lg-W a-s-

\<LN Z

if N is big enough, which in turn implies the Theorem.

LEMMA 13.6 Let

Mk = max{5(pJk), S{pk + 1),..., S{pk+l)} {k = 0,1,... ,n- 1)

and let 0 < M1:n < X2:n < • • • < -Mn.n = M+(pn) be the ordered sample obtained

from the sample Mo, Mi,..., Mn-i- Then for any 0 < e < 1 we have

Mn:n - Mn:n-i > ne a.s.


Proof. Let

An = An(a,e) =

30 < MA = ^°*^/

and for any i,j,m with n(logn)~a < m < n(logn)a,0 < i ^ j < n we have

(lognJa\i- MA <ne\Mj = m}<O[ ne

v & '= O

31 -

\ n2

160 CHAPTER 13

\\~/- j

^

V nl~e )'Let T be a positive integer with T(l — e) > 1. Then only finitely many of the

events Anr will occur with probability 1. Let nT < N < (n + 1)T. Then

AN Ci4(n+i)TBa,e).Consequently only finitely many of the events An will occur with probability 1.

Since

n(logn)~3 < Mn:n < n(lognK a.s.

(cf. the LIL, the Other LIL and Theorem 11.6) we obtain the Lemma.

Lemma 13.6 and Theorem 13.22 combined imply

THEOREM 13.25 For any C > 0 there exists a D = D(C) > 0 such that

sup ?(M+(n) — j,n) < Dlog3n a.s.

j<Clogn


In this section as well as in Section 13.3 we investigated the local time of bigvalues. Many efforts were devoted to studying the local time of small values.

Here we mention only the following:

THEOREM 13.26 (Foldes - Puri, 1989). Let

pN = min{k : \Sk\ = N}

and

E{{-ocN, (*N),pn) = H^:0<k<pN, \Sk\ < ocN}.Then for any 0 < a < 1 we have

limsupfc^»MM = 1 „., A3.13)N-oo 2a2iV2loglogiV

and

liminf5lt^^llM2,oglogiV = 1 .... AS.M)

where Co(ct) is the unique root of the equationa

u tan u =

1-a

in the interval @, |].Note that in case a = l,co(a) = tt/2. Hence A3.13) resp. A3.14) are equivalentwith the Other LIL resp. LIL of Khinchine.

Chapter 14

A few further results

14.1 On the location of the maximum of a

random walk

Let /x(n) (n = 1,2,...) be the location of the maximum of the absolute value of

a random walk {Sn}, i.e. fi(n) is defined by

M{n) = max \Sk\ = 5(/x(n)) and /x(n) < n. A4.1)

If there is more than one integer satisfying A4.1) then the smallest one will be

considered as /x(n). The characterization of the upper classes of /x(n) is trivial,since /x(n) = n i.o. a.s. In order to get some idea about the lower classes we can

argue as follows.

Since lim^oo /x(n) = oo a.s. by the Law of Iterated Logarithm for any e > 0

(n))| < A + e)Bjx(n) log log ^H

with probability 1 if n is big enough. By the Other LIL for any e > 0

consequently

8 log log n

and

BIT

< A+ eJ/x(n) log log /x(n)

7T2 n

161

162 CHAPTER 14

if n is big enough.We ask:

Question 1. Can /x(n) attain the lower bound of A4.2)? The answer is negative.In fact we have

THEOREM 14.1 (Csaki - Foldes - Revesz, 1987).

liminn—>oo

Now we formulate our

Question 2. If

< A "

then by the Law of Iterated

(log log nJn

4 (log log nJ

Logarithm

7T2—

— a.s.

4

for some e > 0 A4.3)

2«) T-,.-," J B log log nI/'/ \ 1/2

. 7T / n \ ,.

2e)-7= :—: . 14.4

We ask: Can | S(n(n)) | attain the upper bound of A4.4) if /x(n) is as small as

possible, i.e. if A4.3) holds? The answer is negative again. In fact we have

THEOREM 14.2 (Csaki - Foldes - Revesz, 1987). Let

Then for any 6 > 0 there exists an e = eF) > 0 such that

2~

A 1/2

_ limsup Mm) < A + 61— a.5.

V n J 2

This theorem roughly says that if /x(n) ~

^fio lognJ 0-e- ^(n) ^s ^^ smaH

possible) then/ \ V2

2 \ log log n

A FEW FURTHER RESULTS 163

Question 3. Intuitively it is clear that M(n) can be (and will be) small if /x(n)is small. Theorem 14.2 somewhat contradicts this feeling. It says that if /x(n) is

as small as possible then M(n) will be small but not as small as possible without

having any condition about /x(n). It will be f (j^^I/2 instead of -^which is the smallest possible value of M(n) by the Other LIL. We ask: How

small can /x(n) be, if M[n) is as small as possible? The answer is:

THEOREM 14.3 (Csaki - Foldes - Revesz, 1987). For any L > 0 there exists

an e = e{L) > 0 such that with probability 1 the inequalities

and

cannot occur simultaneously if n is big enough. However, if g(n) is a positivefunction with g(n) / oo then for almost all u> 6 ft and e > 0 there exists a

sequence 0 < nx = n1(a;,e) < n2 = n2(u;,e) < ... such that

V2

*< g{nk)T. ; rr and hm M(nk) =

-p.'(loglognjkJ »-°°V nk J /8

Question 4. Instead of Question 3 one can ask: How big can /x(n) be, if M(n)is as small as possible? The answer to this question is unknown.

The following theorem gives a joint generalization of the above three theo-

theorems. It also contains the LIL and the Other LIL (cf. Sections 4.4 and 5.3).THEOREM 14.4 (Csaki - Foldes - Revesz, 1987). Let

(log log nJain) = mW,

71

1/2

Then the set of limit points of the sequence (a(n),b(n)) as (n —> oo) is K with

probability 1.

164 CHAPTER 14

Remark 1. This theorem clearly does NOT imply that (a(n),b(n)) 6 K or even

(a(n),b(n)) belongs to a neighbourhood of K if n is big enough. However,

{a(n), b(n)) belongs to a somewhat larger set Ke D K if n is big enough. In fact

we have

THEOREM 14.5 (Csaki - Foldes - Revesz, 1987). Let

Ke = {(*,y) : x > 0,y > 0, ^ + ~ < 1 + e| (e > 0).

T/icn for any e > 0

(a(n),6(n)) €/fg a.s.

/or a// 6uf finitely many n.

In order to formulate a simple consequence of Theorem 14.4 let R*(n) be the

length of the longest flat interval of {M(k),0 < k < n}, i.e. R*(n) is the largestpositive number for which there exists a positive integer a such that

0 < a < a + R* (n) < n

and

Af(a) =M{ot + R*{n)).Then by Theorem 14.1 (or 14.4) we have


. (log log nJ 7T2hminf In — K (nil = —

a.s.n-oo n 4

Equivalently for any e > 0

, 7T2 n

7T2 n

As far as the lower classes of R* (n) are concerned we have


lim.nfloglognn-oo n

'

where 0 is the root of the equationoo okok

= 1

(cf. Theorem 13.3). Equivalently for anys > 0

and

Remark 2. In Theorems 13.1 and 13.3 we investigated the length of the longestflat interval of M+(n). Comparing our results regarding the upper classes we

obtain the intuitively clear fact that the longest flat interval of M+(n) can be

(and will be) longer than that of M(n). Comparing the known results regardingthe lower classes no difference can be obtained.

About the proofs of the above theorems we mention that they are based on

the following:

THEOREM 14.8 (Imhof, 1984). Let ut{x,y) be the joint density of (r^),t~ll2rn{t)) where M(t) is the location of the maximum of a Wiener process. Then

y

14.2 On the location of the last zero

Let ^(n) be the location of the last zero of a random walk {5jk,A: < n}, i.e.

?(n) = max{A: : 0 < k < n, Sk = 0}.

Theorem 13.1 claims that: if g(n) is a nondecreasing sequence of positive numbers

then

if and only if

n=l

166 CHAPTER 14

Consequently for any e > 0

)) and 7T7TI^LUC(tf(n)).(lognJ+e

Since ^(n) = n i.o.a.s. and ^(n) < n the description of the upper classes of

^(n) is trivial.

Here we wish to investigate the properties of the sequence {^(n)} for those

n's only for which Sn is very big or M(n) is very small. It looks very likely that

if Sn is very big (e.g. Sn > BnloglognI/2) then ^(n) is very small. In the

next theorems it turns out that this conjecture is not exactly true. In order to

formulate our results introduce the following notations.

Let f(n) = n1/2jf(n) G ULCEn) with g(n) f oo. Define the infinite random

set of integersZ = Z(f)={n:Sn>f(n)}.

Furthermore, let a(n),/?(n) be sequences of positive numbers satisfying the fol-

following conditions:

a. (n)nonincreasing,

0 < a{n) < 1,

P{n) I 0,

nct(n) | oo, n/3(n) | oo.

Then we have

THEOREM 14.9 (Csaki - Grill, 1988).

na[n) <EUUC(#(n),n€ 2)

if and only if

Further,

nC(n) E LLC(#(n),n € 2)

i/ and only if

oo -I / 2 fn\ \

^(o'jfi') = yz ~02(n)/^2(n) exp (—I < °°-

^in V 2 /

Remark 1. na{n) 6 UUC(#(n),n 6 2) means that na(n) > V(n) a.s. for

all but finitely many such n for which n ? Z. In other words the inequalitiesSn > f{n) and #(n) > na(n) simultaneously hold with probability 1 only for

finitely many n.

In order to illuminate the meaning of the above theorem we present two

examples.

Example 1. Let f(n) = (B - e)n log log nI/2^ < e < 2). Then we obtain that

the inequalities

Sn> (B-e)nloglognI/2 and #(n)>J(l + e)n


Sn > (B - e)nloglognI/2 and V{n)>Ul- e)n i.o. a.s.

The above two statements also follow from Strassen's theorem (Section 8.1).Further,

Sn>{{2- e)nloglognI/2 and #(n) < n(logn)"n i.o. a.s.

if and only if 77 < e. Note the surprising fact that ^f(n) < n(logn)� i.o.

a.s. but there are only finitely many n for which ^(n) < n(logn)� and Sn >

(InloglognI/2 (say) simultaneously hold.

Example 2. Let f(n) = BnloglognI/2. Then we obtain that for any e > 0

the inequalities

Sn > BnloglognI/2 and tf(n) > (- + e) }^^n\2 / log2n


Sn > BnloglognI/2 and ^(n) > --^^ i.o. a.s.2 log2 n

Further,

Sn > BnloglognI/2 and ^(n) < n(loglogn)"f? i.o. a.s.

if and only if 77 < 4.

Now we turn to our second question, i.e.. we intend to study the behaviour of

^ for those n's for which M{n) is very small (nearly equal to 7m1/2 (8 log log n)�/2,

168 CHAPTER 14

cf. the Other LIL E.9)). In this case we can expect that ty(n) is not very small.

The next theorem shows that this feeling is true in some sense. In order to

formulate it introduce the following notations. Let "/(n) and 6(n) be sequences

of positive numbers satisfying the following conditions:

0<-y(n),*(n) < 1,

6(n) nonincreasing, 6(n)n1/2 f oo,

"/(n) monotone, ^{n)n | oo,

))~2 monotone,

o[n)

Then we have

THEOREM 14.10 (Grill, 1987/A).

P{#(n) < ni{n),M(n) < 6{n)n1'2 i.o.} = 0 or 1

depending on whether 73('y,5) < oo or I^jS) = oo where

Consequence. The limit points of the sequence

are the set

{(x,y) : y2 > 4 - 3x,0 < x < 1}.

Example 3. The inequalities

and

/ n \1/2

hold with probability 1 only for finitely many n, i.e. if

/ \ 1/2

then Mln) > D - 3-/ - eW2 —— a.s.

for all but finitely many n. Similarly if

/ \ 1/2

M(n) < D-37-eI/27r -— then tf(n) > -yri a.s.

\81oglogny

for all but finitely many n. This means that if M(n) is very small then ^(n)cannot be too small. For example, choosing D - 3"/ - eI/2 = 1 + 6 we have: if

/ \ 1/2

M(n) <(l+6)ir\ then tf(n) > A - 6)n a.s.v ; ~ v ;

\81oglogn;~


Having the above result we formulate the following conjecture:For any 6 > 0 and for almost all w € ft there exists a sequence of integers

0 < nx = ni(u;,?) < n2 = ri2(u},6) < ... such that

/ \ 1/2

Iand *(nO = nt (t = l,2,...).O<(l + tyr\ 8 log log rii)

The proofs of the above two theorems of this paragraph are based on the eval-

evaluation of the joint distribution of ^(n) and Sn. Here we present such a result

formulated to Wiener process.

THEOREM 14.11 (Csaki - Grill, 1988). Let x > 0,0 < y < 1. Then

> ty,W(t) > xt1'2} = ^,y»;, exp \-where ip(t) is the last zero of W(s),0 < s <t, i.e.

ip{t) = sup{s : 0 < s < t,W(s) = 0}.

2A -y)

14.3 The Ornstein - Uhlenbeck process and a

theorem of Darling and Erdos

Consider the Gaussian process {V(t) = t~ll2W{t)\Q < t < oo}. Then EF(t) =

0, EF2(t) = 1 and EF(t)F(s) = \fs~ft,s < t. The form of this covariance

function immediately suggests that, in order to get a stationary Gaussian process

out of V(t), we should consider

Ua(t) = V(eat), -oo<t<+oo {a fixed > 0).

This latter process is a stationary Gaussian process, for 1&Ua(t)Ua(s) = e~a^~^'2,and it is called Orstein - Uhlenbeck process. We will use the notation U(t) =

f/2(t), and mention, without proof, the following:

170 CHAPTER 14

THEOREM 14.12 (Darling- Erdos, 1956).

lim P{ sup U(t) < a{y,T)} = exp(-e-tf), A4.5)T—oo 0<t<T

lim P{ sup \U(t)\ < a{y,T) = exp{-2e~v), A4.6)T—>oo 0<t<T

where for any —oo < y < oo

a{y,T) = (y + 21ogT + i log log T - ^ log*) BlogT)-1/2.

We also mention a large deviation type theorem of Quails and Watanabe A972)(cf. also Bickel - Rosenblatt, 1973).

THEOREM 14.13 For any T > 0 we have

'/''

= Lz-»oo J 0<KT

Applying their invariance-principle method Darling and Erdos A956) also proved

THEOREM 14.14

and

Jjrn P{max AT^S* < a{y, logn)} = exp(-e-tf)

lim P{max Ar~1/2 | Sk |< a(y,logn)} = exp(-2c-tf)n—> oo 1 < Jk < n

for any -oo < y < oo.

A strong characterization of the behaviour of maxi<jk<n k~l/2Sk is given in the

following:

THEOREM 14.15

max1<Jk<nA: Sk,lim —;

^-=

rrjz—= 1 CIS.

n-oo B log log nI/2

Proof. The LIL of Khinchine implies that

maxi<Jk<n k lSklimsup—;—-=-^r tttt— < 1 a.s.

n-oo Bl0gl0gTlI/2-

Applying Theorem 5.3 we obtain that for any n big enough with probability 1

there exists a

with

such that

Hence

> K1-

max -^ > —?= > \l2\og\ogK, > v^loglogn1"*" > (l - e) \J2 log log nl<*<n y/k y/K

V

and we have Theorem 14.15.

It looks also interesting to study the limit behaviour of the sequence

Ln = max max AT1/2(Si+Jk - S,) (n = 1,2,...).0<j<n l<Jfc<n-j

We prove

THEOREM 14.16

l<liminf——\r-jz < limsup 7——\rjz = K < 00 a.s.n->0° BlognI/2 n_>0O BlognI/2

where the exact value of K is unknown.

Proof. Let an = [(log n)a] (a > l). Then by Theorem 7.13 (see also Section 7.3)for any e > 0 we have

Ln> B logn)�/^ max a^S,-^ - 5,)) > 1 - e a.s.

B log n) 1/2

which proves the lower part of the Theorem.

In order to prove its upper part the following result of Hanson and Russo

A983/A E.11 a)) will be utilized:

If an = f(n) logn with f(n) / 00 then

limsup sup sup , }—> ,,

} < 1 a.s.

n—00 0<j<nan<k<n-j B1ognj1/^

hmsup sup sup ,= < 1 a.s.

n—oo 0<j<n21ogn<Jk<an 2alOgn

hmsup sup sup ,= < 1 a.s.

n-»oo 0<j<n l<Jk<21ogn /2k lOgU

172 CHAPTER 14

Applying again Theorem 7.13 we obtain

k-lt*{Si+k - Sj) ^

hmsup sup sup j====—<

n—oo 0<j<n21ogn<fc<an W2/(n) log n

and clearly

Since f(n) may converge to infinity arbitrarily slowly we obtain the Theorem

by the Zero-One Law.

We present on the value of K of Theorem 14.16 the following:

Conjecture. K = 1.

Let u{n) = u(n,S) resp. u(T) = u(T,W) be the smallest integer resp. the

smallest positive real number for which

resp.

It looks an interesting question to characterize the properties of v(n,S) resp.

u(T,W). Clearly

u(n,S)=n resp. u{T,W) = T i.o. a.s.

On LLC(i/(-,-)) we have


exp({\ognI-') €LLC{u{n,-)).

Proof. By Theorem 14.15 and the LIL of Khinchine we have

which implies Theorem 14.17.

Remark 1. Since the Invariance Principle (cf. Section 6.2) only implies that

max Jfc�/25jk - max r1/2W(*)| < Oil) a.s.,l<*<n l<t<n

\ j\ - \ j

we cannot get Theorem 14.14 from Theorem 14.12 by the Invariance Principle.However, applying Theorem 14.17 we obtain

max k~l'2Sk - max rl^W(t)\ < O (expHlognI"8)) a.s.Kk<n

for any e > 0. Hence we obtain Theorem 14.14 via Theorem 14.12 and the

Invariance Principle.Studying the strong behaviour of U(t) Quails and Watanabe A972) proved


and

14.4 A discrete version of the Ito formula

Ito A942) defined and studied the so-called Itd-integral

f f(W(s))dW(s)Jo

where /(•) is a continuously differentiate function. Here we do not give the

definition but we mention an important property of this integral, the celebrated

ITO-FORMULA (Ito, 1942).

I f(W(s))dW(s) = / f(X)d\ - I ^JO Jo Jo

-ds. A4.7)

In fact A4.7) is a special case of the so-called Ito-formula. Here we are interested

to find the analogue of A4.7) for random walk. In fact we prove

THEOREM 14.19 (Szabados, 1989). Let f{k)(k € Z1) be an arbitrary func-function and define

g(k) = 0

if *>x,

if k = 0,

j=k+l

174 CHAPTER 14

Then for any n = 0,1,2,... we have

i=0 t=0

Remark 1. The function g(-) can be considered as the discrete analogue of the

integral /on f{X)d\, X-+\{f{Si+i) - f{Si)) is the natural discrete version of /'($)and YX=i f{Si)Xi+i can be considered as the discrete Itd-integral.

Proof of Theorem 14.19. We get

(In order to check A4.9) consider the six cases corresponding to 5, = 0, S; >

0,Si < 0;Xi+i = l,Xi-i = -1 separately.) Summing up A4.9) from 0 to n we

obtain A4.8).

Example 1. Let f(t) = t. Then by A4.7) we have

W{S)dW{S) = V^ft-t- A4.10)

and by A4.8)n

ri 4- 1 ^^ «

SiXi+l = g{Z>n+1)— = —

-. A4.HJi=0

ILL

A4.10) and A4.11) completely agree.

Example 2. Let f(t) = i1. Then by A4.7) we have

[tW2(s)dW{s) = ^-^- - ftW{s)ds A4.12)Jo 3 Jo

and by A4.8)

E SfXi+l = g(Sn+l) - E S< - %i = % _ ? Si _ ^ti. A4.13)t=0 t=0

* 6t=0

The term —5n+1/3 of A4.13) is not expected by A4.12). However, we know that

the orders of magnitude of the terms E?=o s<Xi+i, 5^+1 /3 and ?t?=o S{ are n3/2,while that of 5n+i is n1/2 only.

Remark 2. Applying the Invariance Principle 1 A4.7) can be obtained as a

consequence of A4.8).

The celebrated Tanaka formula gives a representation of the local time of a

Wiener process via It6-integral.

TANAKA FORMULA (cf. McKean, 1969). For any x 6 R1 and t > 0 we

have

rj{x,t) = \W(t) -x\-\x\- fts\gn{W{s)-x)dW(s).Jo

Here we are interested in giving the discrete analogue of this formula. In order to

do so instead of f (•, •) we consider a slightly modified version of the local time.

Let

?*(z,n) = #{A: : 0 < k < n,Sk = x).

Then we have

THEOREM 14.20 (Csorgo - Revesz, 1985). For any x € Z1 and n = 1,2,...

n-l

?(x,n) = \Sn -x\-\x\-Y, signEfc - x)Xk+l. A4.14)Jk=O

Proof. Observe that

I — 1 if x = 0,

signEfc -x)Xk+i = -1 (i = l,2,...),

signEfc - x)Xk+i = \Sn - x\ - 1

where

fi=iC{x,n) if x^O,

The above three equations easily imply A4.14).

Remark 3. The Tanaka formula can be proved from A4.14) using Invariance

Principle 1.

Jk=O

n-l

Chapter 15

Summary of Part I

Exact

distr.

Limit

distr.

Upperclasses

Lower

classes

Strassen

type theorems

Sn Th. 2.1 Th.'s

2.9, 2.10

EFKP LIL

Sect. 5.2

Th. 5.1 Strassen's

Th. 1. Sect. 8.1

Mn Th. 2.6 Th. 2.13 EFKP LIL

Sect. 5.2

Th. of

ChungSect. 5.3

Th. 8.2;Wichura's

Theorem

Sect. 8.4

Th. 2.4 Th. 2.12 EFKP LIL

Sect. 5.2

Th. of

Hirsch

Sect. 5.3

Th. 8.2

Th. 9.3 (9.11) Th. 11.1 Th. 11.1 Th. 11.16

The limit behaviour of f@,n) is the same as

that of M+ by Th. 10.3

Th. 9.3 | (9.9) [ Th. 11.6 | Th. 11.6 | Th. 11.17

Since f @,pn) = n a description of f@,n) gives a

description of pn

Z{x,n) Th. 9.4 The limit behaviour is

the same as that of

f@,n) for fixed x,n —> oo.

Th. of

Donsker-

Varadhan

Sect. 11.4

Th. 9.5 The limit behaviour is the same as that of

f@,n)/2 (cf. A0.14)) or M+/2 (cf. Th. 10.4).

177


178 CHAPTER 15

*W

/x(n)

X(n)

Exact

distr.

Th. 9.8

Th. 9.9

Th. 9.10

Th. 13.23

Th. 2.7

Limit

distr.

Th. 9.14, A0.6)(9.12)(9.12)

The limit

tl

(9.12)

Th. 2.14

Upperclasses

Th. 11.5

Th. 13.1

Trivial

Th. 13.1

Th. 13.1

sehaviour ol

lat of R{n)Th. 14.6

Trivial

Trivial

Th. 13.24

Th. 7.2

Lower

classes

Th. 11.4

Th. 13.2

Th. 13.1

Th. 13.3

Th. 13.3'

R{n) is the

by Th. 10.3

Th. 14.7

Th. 14.1

Th. 13.1

Th. 13.24

Th. 7.3

Strassen

type theorems

same as

Replacing Sn, Mn,M+, ?@, n)... by W(t),m(t),m+(t),rj@, t),... respectively the

above-mentioned results remain true by the Invariance Principle 1 (cf. Section

6.3) and Theorem 10.1, with the exception that there is no immediate analogueof 0(n) and the natural analogue of x{n) does n°t have any interest.

In some cases we also investigated the joint behaviour of the r.v.'s of the

above table. A table for these results is

Mn

AC

ACTh.'s 2.5, 2.6,5.8

M~

Th.'s 2.5, 2.6,5.8

Th.'s 5.5, 5.6

Th.'s 14.4, 14.5

14.8

tf(n)Th. 14.9

Th.'s 14.10, 14.11

Clearly many of the results of Part I are not included in the above two tables.

For example, the results about increments, the rate of convergence of Strassen-

type theorems, the results on the stability of the local time, etc. are missing from

the above tables. A summary on the increments of the Wiener process is givenat the end of Section 7.2.


"The earth was without form and void,and darkness was upon the face of the

deep."

The First Book of Moses

II. SIMPLE SYMMETRIC

RANDOM WALK IN ZdNON-ACTIVATEDVERSION

www.avs4you.com

Notations

1. Consider a random walk on the lattice Zd. This means that if the movingparticle is in x ? Zd in the moment n then at the moment n + 1 the

particle can move with equal probabilities to any of the 2d neighbours of

x independently of how the particle achieved x. (The neighbours of an

x ? Zd are those elements of Zd whose d — 1 coordinates coincide with

the corresponding coordinates of x and one coordinate differs by +1 or — 1

from the corresponding coordinate of x.)Let Sn = S(n) be the location of the particle after n steps (i.e. in the

momemt n) and assume that So = 0. Equivalently: Sn = X\ + X2 + • • • +

Xn(n = 1,2,...) where Xx, X2,... is a sequence of independent, identicallydistributed random vectors with

= e,} = V{XX = -e,} = ^ (i = 1,2,... d)

where t\,t2,...,td are the orthogonal unit-vectors of Zd.

d

2. For any x = (xu x2,..., xd) e Rd let ||i||2 = ?>,2.

3. Mn = M(n)= max ||5,||.

4. ?(x, n) = #{fc : 0 < k < n, Sk = x} (x 6 Zd, n = 1,2,...).

5. f(n) = maxf(z,n).

6. px = min{/c : k > 0, Sk = 0},pi = min{/c: k> pi,Sk = 0},

pn = m\n{k : k > pn-i,Sk = 0}.

7. p2k = P{S2k = 0}, p2k+1 =0 (k = 0,1,-2,...).

181

182 II. SIMPLE SYMMETRIC RANDOM WALK IN Zd

8. q2k='P{S2k=0,S2k-2 ^0,...,54^0,52^0}=P{p1 = 2k} =

P{?@,2*)=l, f@,2A:-l)=0} (A: = 1,2,...)-

9- In = P{52 ± 0,54 ± 0,... ,S2Jk_2 ^ 0} = P{e@,2A: - 2) = 0} =

fan (fc = 2,3,...) and i2 = 1.

10. -y = Jlm-y^ = P{nlim f@,n) = 0} = P{Pl = oo} = 1 -

oo n-oo,= i

11. Let W(t) = {Wl{tIW2{tI...1Wd{t)IvrhereWl{tIWt(tI...1Wd[t) are in-

independent Wiener processes. Then the Rd valued process W(t) is called a

d-dimensional Wiener process.

12. m{t) =


Chapter 16

Recurrence theorem

This chapter is devoted to proving the

RECURRENCE THEOREM (Polya, 1921).

PI9 =0¦ \=l1 if d~2'

*¦ n*" '*

[0 if d > 2.

This Theorem was proved for d = 1 in Section 3.1. Hence we concentrate on

the case d > 2.

LEMMA 16.1 For any n = 1,2,...; d = 1,2,...

B»)!= 0} =

nd)

Proof is trivial by a combinatorial argument.

LEMMA 16.2 For any d= 1,2,... as n —> oo we have

( d \d/2P{52n = 0}.« 2 (_j .

Further, in case d = 2

183


184 CHAPTER 16

Proof can be obtained by the Stirling formula.

For later reference we give the following analogues of Lemmas 16.1 and 16.2.

LEMMA 16.3 Let d = 2. Then

2n 2n-k

keAn(x,y)

provided that

where

x + y= 0 (mod 2) and \x\ + \y\ < 2n

keAn(x,y) if and only if k = x (mod 2) and \x\ < k < 2n - \y\.

Proof is trivial by a combinatorial argument.

LEMMA 16.4 (Erdos - Taylor, 1960/A, B.9) and B.10)). Let d = 2,x = (xi,x2) and xx + i2 = 0 (mod 2). Then

P{S2n = x}7TM

if n>\\x\\\

<(-L+O(n-2))exp(-H!) if n<

Proof can be obtained by the Stirling formula.

Similarly one can obtain

LEMMA 16.5 Let d > 2, x = (xx,x2,.. .,xd) and xx + x2 + ••• + xd

(mod 2). Then

= 0

P{S2n =

LEMMA 16.6 In case d = 2

2n

if n>\\x\\\

if n<\\x\\\

lim — = 1.

k=l

RECURRENCE THEOREM 185

Proof. Clearly

E E p{^i = 0, S2k = 0} = 2 ?2 P{S2j = 0,S2j+2fc = 0} + ? p{52i = 0}

= 2

2 ^

Hence we have Lemma 16.6.

Proof 1 of the Recurrence Theorem. In the case d = 2 our statement follows

from Lemmas 16.2, 16.6 and Borel - Cantelli lemma 2* of Section 4.1, while in

the case d > 3 it follows from Lemma 16.2 and Borel - Cantelli lemma 1 (cf.Section 4.1).

Remark 1. Lemma 16.6 is also true in the case d = 1. (The proof is essentiallythe same.) Hence we obtain a new proof of the recurrence theorem in the case

d = 1. The third proof (cf. Section 3.2) applied in case d — 1 does not work in

the case d > 2. The idea of the first proof can be applied in the case d = 2 but

it requires hard work. The second proof can be used without any difficulty.

Proof 2 of the Recurrence Theorem. Introduce the following notations:

Po = l, P2k = P{S2k =

O}*2y4k^j ,

q2k = P{S2k = 0,52Jk_2 ^ 0,52Jk_4 ^ 0,..., S2 ^ 0},

P(z) = f>2fcz2\ Q{z) = fjq2kz2k.

Between the sequences {p2jt} and {<72jt} one can easily see the following relations:

@) po = 1,

A) p2 = q2,

(ii) p\ = <74 "H qiPii

(iii) p6 = q6 + q4p2-

(k) p2jfc = <72Jfc + <72Jfc-2P2 H h <72P2Jfc-2>

186 CHAPTER 16

Multiplying the k-th equation by z2k and adding them up to infinity we obtain

P(z)=P{z)Q{z) + l i.e. Q(z) = l-J^. A6.1)

By Lemma 16.2 we obtain

hm P(z) {= oo if d < 2,< oo if 3,

i.e.

if d < 2,if 3.

Since Q(l) = Yl<?=iQ2k is the probability that the particle eventually visits the

origin we obtain

oo \~l

< 1 if d > 3,

"=1

[I if d = 2.

Hence we have the theorem.

Remark 2. In the case d > 3 the formula

P{5n = 0 for some n = 1,2...} = 1 - P{ lim f@, n) = 0}

l-l^— A6.2)

Jk=O

is applicable to the evaluation of the probability that the particle returns to the

origin. In fact by Lemma 16.2 we have

Jk=O Jk=n Jk=O

where Y%ZoP2k can be numerically evaluated by Lemma 16.1. For example, in

the case d = 3 one can obtain Z)*Li 92* ~ 0.35. In fact this method is not

easily applicable for concrete calculations. Griffin A989) gave a better version

of it and evaluated the probability of recurrence for many values of d. For

example, for d = 3,4,20 he calculated that the probabilities of recurrence are

RECURRENCE THEOREM 187

0.340537, 0.193202, 0.026376. Note that in the case d = 20, P{52 = 0} = 0.025.

Hence the probability that the particle returns to the origin but not in the second

step is 0.001376.

It looks also- interesting to estimate the probability

n-l

that the particle returns to the origin but not in the first 2n - 2 steps. We prove

LEMMA 16.7 (Dvoretzky - Erdos, 1950). For any d > 3

>2n ~ 2) = °> ~

1

A6.3)k=rt

Proof. Classifying the paths according to the last return to the origin, we get

n

2.= °'5i 5* °'.i = 2i + l> + 2,... ,2n}

t=0

n

EPE2. = °'5i " S* * °5i = 2i + l>2i + 2,... ,2n}i=0

n

Subtractingn

EP2.^« = ^— A6.5)

i=0

from both sides of {16.4) we have

A6-6)

Since q2n \ 1 we have

n

1 - An > l2n+2- An + -y J2 Pn- (I6-7)


188 CHAPTER 16

Since

)t=l \t=0 /

(cf. 10 of Notations and A6.2)) we obtain (by A6.5), A6.8), A6.7) and 9 of

Notations)

P*i = l~An> l2n+2~ An +

t=0 t'=0

n n

= l2n+2 -lJ2PK+iY2 PK = "t**+* ~^=t=0 t=l j=n+l

Consequently (by A6.8))

. A6-9)

t=0

Hence we have A6.3) by Lemma 16.2.NON-ACTIVATEDVERSIONwww.avs4you.com

Chapter 17

Wiener process and Invariance

Principle

Let Wx(t), W2(t),..., Wd{i) be a sequence of independent Wiener processes. The

Rd valued process W(t) = {Wx(t),W2{i),. ..,Wd{i)} is called a d-dimensional

Wiener process. We ask how the random walk Sn can be approximated by H^(t).The situation is very simple if d = 2.

Consider a new coordinate system with axes y = x, y = — x. In this coordinate

system

Xn = A,0) in the original system ,

Xn = @,1) in the original system ,

xn = (-1,0) in the original system ,

xn = @,-1) in the original system .

Observe that the coordinates of Xn are independent r.v.'s in the new coordinate

system (it is not so in the old one); hence by Invariance Principle 1 of Section

6.3 there exist two independent Wiener processes W\{t) and W2(t) such that

'

B-V2,(-2�/(-2�/

k B-1'2,

2-1/2)2? 2-1/2)2?_2-l/2)_2-i/2)

if

if

if

if

\2-1/2Wx (n) - S™ | = O(log n) a.s.

and

a.s.

where SW,S$*) are the independent coordinates of Sn in the new coordinate

system. Consequently we have

THEOREM 17.1 Let d = 2. Then on a rich enough probability space {fl, 7,P}one can define a Wiener process W(-) ? R2 and a random walk Sn € Z2 such

that.i _ i/•> i \ii —/• \

a.s.

189


190 CHAPTER 17

In the case d > 2 the above idea does not work. Instead we define

/C?) = #{k : 1 < k < n,Xk = e, or - e,} (t = 1,2,... ,d)

where e< is the t'-th unit vector in Zd. Then by the LIL we have

a.s. A7.1)

for any e > 0 and for all but finitely many n.

Let Sn = (SJil\SJ?\...,SJ4) (where SW is the i-th coordinate of Sn in the

original coordinate system). Then by Invariance Principle 1 there exist indepen-independent Wiener processes WX(-),W2{-),...,Wd{-) such that

= O(log /CW) = O(logn) a.s.

for any i = 1,2,..., d. By A7.1) and Theorem 7.13 we have

a.s.

Consequently we have

THEOREM 17.2 On a rich enough probability space {fi,7,P} one can definea Wiener process W(-) ? Rd and a random walk Sn 6 Zd such that

~ W il logn

for any d = 1,2,

It is not hard to prove that

P{W(t) = 0 i.o.} =1 if d = 1

and

P{W(t) = 0 for any t > 0} = 0 if d > 2.

Hence we can say that the Wiener process is not recurrent if d > 2. However, it

turns out that it is recurrent in a weaker sense if d = 2.

THEOREM 17.3 (see e.g. Knight, 1981, Th. 4.3.8). For any

e > 0 we have

P{\\W(t)\\<e t.o.} = l •/ d = 2, A7.2)

P{|lwr(*)ll < e- i.o.} = 0 if d>Z A7.3)where i.o. means that there exists a random sequence 0 < t\ = ti(u>,e) < t2 =

t2{u,e) < ... such that limn_oo tn = oo a.s. and ||W(tn)|| < e (n = 1,2,...).

The proof of Theorem 17.3 is very simple having the following deep lemma

which is the analogue of Lemma 3.1.

LEMMA 17.1 (Knight, 1981, Th. 4.3.8). Let 0 < a 0, \\W(t + s)\\ = a} < inf{s : s > 0, \\W{t + s)\\ = c} \ B}

' c-b

c — a

logc-

logc -

c*-d-

log 6

log a

b2-<t

if

if

if

d =

Au —

d>

1,

2,

3,(> ci~a - a'

where B = {\\W{t)\\ =6}.

Remark 1. A7.3) is equivalent to

lim ||W(i)|| = oo a.s. if d > 3. A7.4)t-»oo

The rate of convergence in A7.4) will be studied in Chapter 18.

In connection with A7.2) it is natural to ask how the set of those functions

et can be characterized for which

T{\\W{t)\\<et i.o.} = l. A7.5)

This question was studied by Spitzer A958), who proved

THEOREM 17.4 Let g(t) be a positive nonincreasing function. Then

g(t)t^eLLC(\\W(t)\\) (d = 2)

if and only if ?~=1(fc | log </(*:) I)� < oo.

Remark 2. Theorem 17.4 implies

(log0. fLLC(||W(t)||) if e>0,1 e

\)) if ?<0.

The proof of Theorem 17.4 is based on the following:

192 CHAPTER 17

LEMMA 17.2 (Spitzer, 1958). For any 0 < tx < t2 < oo we have

Here we also mention a simple consequence of Theorems 2.12 and 2.13 (cf.also Theorem 6.3).

THEOREM 17.5 For any d= 1,2,... and T > 0 we have

P(m(T) > uT1/2) = O(trVu3/2) as u -> oo

and

P(m(T) < uT1'2) = exp(-O(u-2)) as u -> 0.

Similarly for any d= 1,2,... as N —> oo we have

?{M{N) > uN1'2) = exp(-O(u2)) if u -> oo but u < Nx'z

and

Y{M{N) < uNxl2) = exp(-O(u�)) if u -> 0 6u* u > iV�/3.NON-ACTIVATEDVERSION

www.avs4you.com

Chapter 18

The Law of Iterated Logarithm

At first we present the analogue of the LIL of Khinchine of Section 4.4.

THEOREM 18.1

\imsnpbt\\W{t)\\ = l a.s. A8.1)t-»oo

and

limsup6n<i1/2||Sn|| = 1 a.s. A8.2)n—»oo

where bt = B* log log*)�/2.

Proof. By the LIL of Khinchine we obtain

> 1.*¦ -

rr \ / r r—

t-»oo

In order to obtain the upper estimate assume that there exists an e > 0 such

that

limsupbt\\W{t)|| > 1 + e: a.s.

t-»oo

(Zero-One Law (cf. Section 3.2)). For the sake of simplicity let d = 2 and define

4>k = karccos0,(k = 0,1,2,... ^(arccos©)�]) where 0 = A + er/2)(l + er)�.If 6t||VV(t)|| > 1 + e then there exists a k such that

bt\cos<j>kWx(t) +sxn<{>kW2{t)\ > 1 + |. A8.3)

Since cos 4>kWi{i) + sin<f>kWi(t){t > 0) is a Wiener process A8.3) cannot occur if

t is big enough. Hence we have A8.1) in the case d = 2. The proof of A8.1) for

d > 3 is essentially the same. A8.2) follows from A8.1) by Theorem 17.2.

Applying the method of proof of Strassen's theorem 2 of Section 8.1 we obtain

the following stronger theorem:

193

194 CHAPTER 18

THEOREM 18.2 The process {btW(t),t > 0} is relatively compact in Rd with

probability 1 and the set of its limit points is

Cd = {xeRd,\\x\\<\}.

The real analogue of Strassen's theorem can also be easily proved. It goes

like this:

THEOREM 18.3 The net {btW(xt),0 < x < 1} is relatively compact in

C@,1) x • • • x C@,1) = (C@, l))d and the set of its limit points is S* where Sd con-

consists of those and only those Rd valued functions f(x) = (fi{x), fi{x),..., fd{x))for which /,@) = 0,/,-(•)(i = 1,2...,d) is absolutely continuous in [0,1] and

We ask about the analogue of the EFKP LIL of Section 5.2. It is trivial to

see that if a(t) ? ULC{H^(t)} in the case d = 'l then the same is true for any d.

However, the analogue statement for UUC{Wr(t)} is not true. As an example,we mention that Consequence 1 of Section 5.2 tells us that in the case d = 1 for

any e > 0

Sn < [2n Hog log n + ( - + e) log log log n) J1/2

a.s.

for all but finitely many n. However, it turns out that in case d > 1 it is not

true. In fact for any d > 1

l l l ) J i.o. a.s.d1/2Sn > Bn Hog log n H —- log log log n) J

Now we formulate the general

THEOREM 18.4 (Orey - Pruitt, 1973). Let a(t) be a nonnegative nondecrtas-

ing continuous function. Then for any d= 1,2,...

tl'*a[t)€WC[\\W[t)\\)

and

if and only if

r feJ\

THE LAW OF ITERATED LOGARITHM 195

Remark 1. The function

n

does not satisfy A8.4) if a2 = 2,a3 = d + 2,ak = 2 for 4 < k < n but it does if

an is increased by e > 0 for any n > 2.

It was already mentioned (Chapter 17, Remark 1) that

lim 11^@11 = °° a.s. if d > 3. A8.5)t—>OO

Now we are interested in the rate of convergence in A8.5). This rate is called

rate of escape. We present

THEOREM 18.5 (Dvoretzky - Erdos, 1950). Leta(t) be a nonincreasing, non-

negative function. Then

tl'*a{t)eLLC(\\W(t)\\) (d>3)

and

n^ofrOeLLC^HSnll) {d > 3)

if and only if

f>B"))«-2 < oo. A8.6)n=l


a(x) = (logxlog2 x... (logfcxI+eJdoes not satisfy A8.6) if e < 0, but it does if e > 0.

In the case d = 2 we might ask for the analogue of Theorem of Chung of

Section 5.3, i.e. we are interested in the liminf properties of

m(t) = sup \\W{s)\\ and M{n) = max ||5fc||.

This question seems to be unsolved.

Theorems 18.4 and 18.5 together imply: there are infinitely many n for which

\\Sn\\ > d

and for every n big enough

||S»|| > n^logn)—a±i [d > 3, e > 0). A8.8)

Erdos and Taylor A960/A) proved that if a particle is very far away from the

origin, i.e. A8.7) holds, then it may remain far away forever (d > 3). In fact we

have the following:

196 CHAPTER 18

THEOREM 18.6

k>n


Chapter 19

Local time

19.1 ?@,n)inZ2The Recurrence Theorem of Chapter 16 clearly implies

n) {- '

hm f@,n) { ..

n-»oosv '

y < oo if a > 3.

Hence we study the limit properties of f@, n) in the case d = 2.

THEOREM 19.1 (Erdos - Taylor, 1960/A). Letd = 2. Then

lim P{f@, n) < x log n} = 1 - e"™

uniformly for 0 < x < (log nK/4 and

()

The proof of this theorem is based on the following:

LEMMA 19.1 (Dvoretzky - Erdos 1950, Erdos - Taylor 1960/A).

>n} = P{?@,n) = 0} = ^ + O((logn)�) (d = 2).

Proof. By A6.4) (cf. also 9. of Notations) we have

n-l

?P{S2fc = 0}P{?@,2n-2A:-2) = 0} = l (n = l,2,...) A9.1)Jk=O

197

198 CHAPTER 19

where ?@,0) = 0. Since by Lemma 16.2

P{S2Jk = 0} « (fcTr)�

we have

E !^. A9.2)Jk=O

W

Since the sequence P{?@, 2n) = 0} is nonincreasing by A9.1) and A9.2) we

obtain

1 > P{?@,2n - 2) = 0} ]TP{S2fc = 0} « P{e@,2n - 2) =

*=o"

and

P{e@,2n)=0}<^^. A9.3)

Similarly for any 0 < k < n by A9.1)

/ k \ n~l

1 < EPfe = 0) P(e@,2n - 2k - 2) = 0) + ? PE2j = 0). A9.4)

Thus, if k tends to infinity together with n, A9.4) yields

7T 7T

Taking k = n — [n/ log n] we have

P{?@,2n)=0}>logn

Hence we have the main term of Lemma 19.1. The remainder can be obtained

similarly but with a more tedious calculation.

Proof of Theorem 19.1. Let q = [xlogn] + 1 and p = [n/q\. Then

, n) > xlog n} > P{pq < n} > f[ P Lk -

Pk-X < R |

Assuming that x < logn(log2 n)~l~e by Lemma 19.1 we obtain

P{f@,n) > xlogn} >e-"(l+o(l)) (n-> oo). A9.5)

LOCAL TIME 199

In fact we obtain that

P{?@,n) > xlogn} > e~rx{\ + o((logn)-1/4))

uniformly in x for x < (lognK/4.In order to get an upper estimate, let Ek(k = 1,2,... ,q) be the event that

precisely k of the variables p,—

p<_i (i = 1,2,..., q) are greater than or equal to

n, while q— k of them are less than n. Then

n) > xlogn} C

k=l

Hence

O(log nJ

by Lemma 19.1 uniformly in x for x < (lognK/4.A9.5) and A9.6) combined imply the first statement of Theorem 19.1. The

second statement can be obtained similarly observing that by A9.6) and Lemma

19.1 for any x = xra we have

P{f@,n) > xlogn} < exp (-irx + O

Note that Theorem 19.1 easily implies

THEOREM 19.2 Let d = 2. Then

Jim P |pn < exp [-) | = exp(-7rZ)

uniformly for 0 < z < n3/7.

Clearly for any fixed x E Z2 the limit properties of ?(x,n) are the same

as those of f@,n). For example, Theorem 19.1 and Lemma 19.1 remain true

replacing f@,n) by f (x, n). However, if instead of a fixed x a sequence xra (with||xn|| —»• oo) is considered the situation will be completely different. The followingresult gives some information about this case.

200 CHAPTER 19

THEOREM 19.3 (Erdos - Taylor, 1960/A). Let d = 2. Then

f 2log||x|| /1 + o/log3|kllogn

if 20 < ||x|| < n1/3,

ilogn

l + O

log.

log

n 1/2

20

Proof. By A6.4) we have

P2»T2n-2»+2 =

i=0

and similarly

,2n) = 0} + ? P{52Jk = x}72n-2Jk+2 = 1

A9.7)

A9.8)Jk=i

provided that x = (xi, x2) with Xi+x2 = 0 (mod 2). A9.7) and A9.8) combined

implyn

,2n) = 0} -Jk=l

Consequently for 1 < kx < k2 < n we get

= 0} - 72n+2 < 72n-2Jk,+2Jk=l

72n-2Jk3+2ik=iki + l

Now in the case 400 < ||x||2 < n2/3 put kxLemmas 19.1, 16.2 and 16.4 we obtain

= \\x\\2 and k2 = [n4/5] then by

P{f(z,2n)=0}-72n+2<>gBn - (log ny)) ? VA:7r

+ °\k*

LOCAL TIME 201

i \\

logBn - 2k2)+ O

x

<iogikirlogn

Similarly, for 1 < k3 < n

,2n) .= 0} - l2n+2 > l2n+2 ?>2Jk - P{52JkJk=i

Take

, I'll*3

~~

i ii ii2"log2 ||x||

This, by Lemmas 16.2, 16.4 and 19.1, implies

hence we have Theorem 19.3 in the case 20 < ||x|| < n1/3. The case n1/6 <

n1/2/20 can be treated similarly.As a trivial consequence of Theorem 19.3 we prove

LEMMA 19.2 Let nk = [exp(e*log/E)]. Then for any k big enough

,nf*) - ?@,n4) = 0 | Sr,j = 0,1,2,... ,nk} <3

log A;

Proof. Since \\Snk\\ < nk, for any x e Z2 with ||x|| = nk we have

I>{Z@,nlko*k)-t@,nk)=0\Sj;j = 0,1,2,. ..,nk}

J -

log*"The next theorem gives a complete description of the strong behaviour of

202 CHAPTER 19

THEOREM 19.4 (Erdos - Taylor, 1960/A). Let d = 2 and let f{x) resp. g{x)be a decreasing resp. increasing function for which

/(x)logx / oo, g{

Then

n.)) A9.9)

if and only if

f 9\x) r-g(x)j ^ fin irj\J\ x log x

and

f{n) log n e LLC(f@, n)) A9.11)

if and only if

l°°J^-dx < oo. A9.12)J\ xlogx


g{x) = log3 x + 2 log4 x + log5 x + ¦ ¦ ¦ + log^ x + t \o%k+1 x

satisfies A9.10) if and only if r > 1. The function

f(x) = (log log x)�"'

satisfies A9.12) if and only if e > 0.

Proof of A9.9). Instead of A9.9) we prove the somewhat weaker statement

only: for any e > 0

A + ejir-^log n) log3 n G UUC(f@, n)) A9.13)

and

A - ^TT-^log^logan 6 ULC(f@,n)). A9.14)

Let njk = [exp((l + e/Z)k)}. Then by Theorem 19.1

P {^@, nk) > (l + |) ^Oog nk) log3 n

and by Borel - Cantelli lemma

, nk) < fl + |J ^(log nk) log3 njk a.s.

LOCAL TIME 203

for all but finitely many k. Let n^ <n < njk+i. Then

f@, n) < f@, Mjk+1) < (l + |^ TT-^log Mjk+1) log3 Mjk+1 < A + ^TT-^log n) log3 M

if k is big enough. Hence we have A9.13).Now we turn to the proof of A9.14). Let

njk =

^Jk = { —r—— > log3 "Jfc

and

Mjk+1

Then clearly Fk C ^7fc+i and by Theorem 19.1 and Lemma 19.2

and similarly for any j < k

Hence we obtain A9.14) by Borel - Cantelli lemma.

Proof of A9.11). Instead of A9.11) we prove the somewhat weaker statement

only: For any e > 0

(lognHloglogn)� e LUC(f@,n)) A9.15)

and

n)-1-' e LLC(f@,n)). A9.16)Let njk = [exp(efc)j. Then by Theorem 19.1

nt)<,. ^(loglognjk

and by Borel - Cantelli lemma

"¦

204 CHAPTER 19

for all but finitely many k. Let nk < n < nk+i. Then

jfe logn>

ifcI+e>

(loglogri)i+2e

if k is big enough. Hence we have A9.16).Now we turn to the proof of A9.15). For any r = 1,2,... we have

exp(Cr2r)} = P{p2'-» < exp(Cr2r)}

= P{<;@,exP(Cr2')) > 2'�} » exp (-Since the r.v.'s p2'

—

P2'-1 are independent we obtain

Pv > p-iT~

Pv~x ^ Cr2r i.o. a.s.

and consequentlyf@,exp(Cr2r)) < 2r i.o. a.s.

Let n = exp(Cr2r) with C = log 2. We get

f@, n) < -—: i.o. a.s.

log log n

Hence the proof of Theorem 19.4 (in fact a slightly weaker version of it) is com-

complete.Note that Theorem 19.4 easily implies

THEOREM 19.5 For any s > 0

exp(n(lognI+e)eUUC(pn),MlognI-') GULC(pn),

) GLUC(pn),

exp (A - e)-^— ) e LLC(pn).

19.2 f (n) in Zd

As we have seen (Recurrence Theorem, Chapter 16)

lim f@, n) < oo a.s. if d > 3.

LOCAL TIME 205

Similarly for any fixed x G Zd

lim ?(x, n) < oo a.s. if d > 3.n—»oo

v

However, it turns out that

THEOREM 19.6 For any d>l we have

lim f (n) = lim sup ?(x, n) = oo a.s.n—oo n—oo ln—oo

Proof. Theorem 7.1 told us that the length Zn of the longest head run till

n is a.s. larger than or equal to A - e) log n/ log 2 for any e > 0 if n is bigenough. Similarly one can show that the sequence X\, X2, ...,Xn contains a run

«i> -«i, «i,—

«i, ••-,«! of size (l — e) log n/ log 2d. This implies that

minf>n-»oo Jog n 2

which, in turn, implies Theorem 19.6.

A more exact result was obtained by Erdos and Taylor A960/A). They proved

THEOREM 19.7 For any d>3

lim,

= \d a.s.n—oo Jog n

where

\d = -(logP{lim ?@,n) > I})� = -(log(l -7))�.n—*oo

In the case d = 2

— < liminf < limsup — < - a.s.

4tT n—oo (lg)-4 (log n)* 7Tn—oo

Erdos and Taylor A960/A) also formulated the following:

Conjecture. For d = 2

, €(») 1lim T rr = - a.s.

n-^°° (lognj2 7T

206 CHAPTER 19

19.3 A few further results

First we give an analogue of Theorem 13.8.

THEOREM 19.8 (Erdos - Taylor, 1960/A).

..1 A _1_ 1

f A ohm 2^ ]= ~ a-5- */ d = 2.

n—oo log nk=x log Pk 7T

The next theorem is an analogue of Theorem 13.1.

THEOREM 19.9 (Erdos - Taylor, 1960/A). Let d = 2,/(n) | oo as (n -> oo)and En be the event that the random walk Sn does not return to the origin between

n and nf^. Then

P(En i.o. ) = 0 or 1

depending on whether the series

oo1

converges or diverges.

Now we turn to the analogue of Theorem 11.27.

THEOREM 19.10 (Erdos - Taylor, 1960/A). Let fr{n) be the number of pointsvisited exactly r times up to n. Then

lim/i(")log'n=1 ^ tf d = 2

and

Jim^-^ =1(l-1)k~1 a.s. if d > 3, k = 1,2,...

where 7 = i(d) is the probability that the path will never return to the origin.

We investigated the favourite values in Section 11.6. We could ask the ana-

analogue questions in the case d > 2. Theorem 19.7 combined with Theorem 19.4

imply that the favourite values are going to infinity in the case d > 2 just like

in the case d = 1 (cf. Theorem 11.26). However, the rate of convergence is not

clear at all. A partial result will be given in Theorem 22.8.


Chapter 20

The range

20.1 The range of Sn

Let R{n) be the number of different vectors among Si,S2,...,Sn, i.e. R(n) is

the number of points visited by the particle during the first n steps. The r.v.

•>

will be called the range ofSi,S2,..., Sn. In the case d = 1 Theorem 5.7 essentiallytells us that R{n) is going to infinity like n1/2. In the case d = 2 Theorems 19.1

and 19.4 suggest that R(n) ~ n(logn)�. (Since any fixed point is visited logntimes the number of points visited at all till n is n(logn)�. Clearly it is not a

proof since some points are visited more frequently (cf. Theorem 19.7) and some

less frequently (cf. Theorem 19.10) among the points visited at all.) In fact we

prove

THEOREM 20.1 (Dvoretzky - Erdos, 1950).

EJE(n) =

nn ( n log log n

logn \ (lognJ

lim ?@, n) = 0

limf@,n) =0

limf@,n) = 0

nP

nP

nP

O(logn)

if d = 2,

if d = Z,

if d = 4,

if d>5

B0.1)

207


208 CHAPTER 20

where Cd (d = 5,6,...) are positive constants and

/V log log n^ .. ,

** ~n \T~ tf d = 2,I V (log"K /

Vari2(n) = Ei22(n) - (Ei2(n)J < { O(n3/2) ,/ d = 3, B0.2)O(n log n) if d = 4,

O(n) if d>5.

Further, the strong law of large numbers

= 1 a.s. if d > 2. B0.3)

Remark 1. Theorem 5.7 implies that the range does not satisfy the strong law

of large numbers in the case d = 1.

The proof of B0.1) is based on the following:

LEMMA 20.1

P{5n ^ Si fori = l,2,...,n-l}= P{f@,n - 1) = 0}. B0.4)

Remark 2. The left hand side of B0.4) is the probability that the n-th steptakes the path to a new point.

Proof of Lemma 20.1.

= P{Xn + Xn_! + • • • + Xi+1 ? 0 for t = 1,2,..., n - 1}= P{X1 + X2 + --- + Xn-i ^ 0 for t = 1,2,... ,n

- 1}

y ^ 0 for j = 1,2,..., n - 1} = P{?@, n - 1) = 0}.

Hence we have B0.4).Let

(l if Sn^Si for t = l,2,...,n-l,Vn

[0 otherwise.

Then

Jk=i

THE RANGE 209

Consequently by Lemma 20.1

EJ2(n) = E ? V* = E PU(°> * - 1) = 0}-Jk=l Jk=l

Hence B0.1) in the case d = 2 follows from Lemma 19.1, and in the case d > 3

it follows from Lemma 16.7.

In order to prove B0.2) we present two lemmas.

LEMMA 20.2 Let 1 < m < n. Then

Proof.

^ 5m, i = 1,..., m - 1; Sj ? Sn, j = 1,..., n -

5m, t = 1,..., m - l}P{5y ^ Sn, j = m,..., n - 1}

Sm, i = 1,..., m - l}P{Sy ^ 5n_m+i, j = 1,... n - m}

LEMMA 20.3

Vari2(n) < 2Ei2(n) (e72 (n - [^]) - EJ2(n)

Proof. By Lemma 20.2

Vari2(n) =

<2 E

2

Since by Lemma 20.1 E^y is nonincreasing the max is attained for t = [n/2] + 1

and Lemma 20.3 is proved.Then B0.2) follows from Lemma 20.3 and B0.1). B0.3) can be obtained by

routine methods.

Donsker and Varadhan A979) were interested in another property of R(n).In fact they investigated the limit behaviour of Eexp(—vR(n))(v > 0, n —> oo).They proved

210 CHAPTER 20

THEOREM 20.2 For any u > 0 and d = 2,3,...

lim n~*+* logE(exp(-i/i2(n)) = -k{u)

where

and at is the lowest eigenvalue of —1/2A for the sphere of unit volume in Rd

with zero boundary values.

Remark 3. In the case d = 2 Theorem 20.1 claims that R{n) is typicallyTrn/logn. Hence we could expect that Eexp(—uR(n)) ~ exp(—i/irn/logn).However, Theorem 20.2 claims that Eexp(—uR{n)) ~ exp(—k{u)nll2). Compar-Comparing these two results it turns out that in the asymptotic behaviour of Ee~"^n^the very small values of R(n) contribute most. This fact is explained by the

following:

LEMMA 20.4 For any v > 0 there exists a Cv > 0 such that

E(exp(-i/JE(n))) > exp(-Cun1/2).

Proof.

E(exp(-i/JE(n))) > E(exp(-i/iE(n)) | Mn < nl'A)Y{Mn < n1/4)> exp(-i/?rn1/2)P(Afn < n1/4).

By Theorem 17.5

P(Mn<n1/4)=exp(-O(n1/2))and we have Lemma 20.4.

20.2 Wiener sausage

Let W(t) € Rd be a Wiener process. Consider the random set

Br{T)=U0<t<T(W{t)+Kr)= {x: xe Rd,x = W(t) + a for some 0 < t < T and a € Kr}

where

Kr = {x: \\x\\ < r}.

Br(T) is called Wiener sausage. The most important results are summarized in

THE RANGE 211

THEOREM 20.3 (cf. Le Gall, 1988).

lim !^A(Br(T)) = 2tt a.s. B0.5)

for any r > 0 and d = 2. If d > 3 then

WmT-lX{Br{T))=c{r,d) a.s. B0.6)T—>oo

where c(r,d) is a positive valued known function of r and d. Further,

lim P{Kd(T)(X(Br(T)) - Ld{T)) < x} = $(ai + 0) B0.7)T—*oo

where

,2tt- if d = 2,

'

l{d,r)T if d>3,

{^P- H d = 2,

(TlogT)�/2 if d = Z,T*~*/2 if d > 4

and a,/?,7 are known functions of r and d.

Clearly B0.5) and B0.6) are strongly related to B0.1) and B0.3). In fact

one can define the random walk sausage as

Jk=O

Then Theorem 20.1 implies

lim l^-\{Bls){N))=ir2r2 a.s. if d = 2N-00 N

and

lim N^XiB^iN)) = r27rP{ lim f@,n) = 0} a.s. if d > 3.

The analogue of B0.7) is unknown for random walk sausage but we can

formulate the following:

212 CHAPTER 20

Conjecture 1.

lim P {K{dS)(N)(X(Bls\N)) - L{fJ(N)) < x} = *N—KX3 (

where

(^^L if d = 2,log AT

r27rP{ lim f@, n) = 0}N if d > 3

and K, a, C are suitable normalizing constants.

Chapter 21

Selfcrossing

It is easy to see (and Theorem 20.1 also implies) that the path of a random walk

crosses itself infinitely many times for any d > 1 with probability 1. We mean

that there exists an infinite sequence {Un,Vn} of positive integer valued r.v.'s

such that S(Un) = S(Un + Vn),a.nd0<U1<U2<...,{n = l,2,...). However,we ask the following question: will selfcrossings occur after a long time? For

example, we ask whether the crossing S(Un) = S(Un + Vn) will occur for everyn = 1,2,... if we assume that Vn converges to infinity with a great speed and Unconverges to infinity much slower. In fact Erdos and Taylor A960/B) proposedthe following two problems.

Problem A. Let f(n) | oo be a positive integer valued function. What are the

conditions on the rate of increase of f(n) which are necessary and sufficient to

ensure that the paths {So, Si,..., Sn} and {?„+/(„), ?„+/(„)+i;...} have points in

common for infinitely many values of n with probability 1?

Problem B. A point Sn of a path is said to be "good" if there are no pointscommon to {S0,Si,...,Sn} and {5rn+1,5rn+2,...}. For d = 1 or 2 there are no

good points with probability 1. For d > 3 there might be some good points: how

many are there?

As far as Problem A is concerned we have

THEOREM 21.1 (Erdos - Taylor, 1960/B). Let f(n) | oo be a positive integervalued function and let En be the event that paths

{S0,Si,...,Sn} and {Srn+/(n)+1,5rn+/(n)+2,...}have points in common. Then

(i) for d = 3, if f(n) = n(<p(n)J and <p(n) is increasing, then

P(En i.o.) = 0 or 1 B1.1)

213

214 CHAPTER 21

depending on whether Efc^i((PBA:)) lconverges or diverges,

(ii) for d — 4, if f(n) = nx{n) and x{n) I5 increasing, then we have B1.1)depending on whether E^=i(^xBA:))� converges or diverges,

(iii) for d> 5, if

supm>n rn n

(for some C > 0) then we have B1.1) depending on whether

n=l

converges or diverges.

An answer of Problem B is

THEOREM 21.2 (Erdos - Taylor, 1960/B). For d>Z let G^(n) be the num-

number of integers r (l < r < n) for which (Sq, Si, ..., Sr) and (Sr+i, Sr+2, ¦ ¦ •) have

no points in common. Then

(i) d = 3. For any e > 0

P{GC)(n) > n1/2+e i.o. } = 0.

(ii) d = 4.

P{0 = liminf" rr""'< limsup- ^)log"< C) = 1.n—»oo fi n—>oo M J

(iii) d > 5.

hm i—- = t& a.s.n—oo n

where Tj is an increasing sequence of positive numbers with r^ f 1 as d —> oo.

Remark 1. Applying Theorem 21.1 for d = 4 and /(n) = n — 1 we find that for

infinitely many n the paths {So, ^i, • • • > Sn} and {^n, 52n+i, • • •} have a point in

common. This statement is not true for d > 5.

Our next theorem is intuitively clear by Remark 1.

THEOREM 21.3 (Erdos - Taylor, 1960/B, Lawler, 1980). For d = 4, two inde-

independent random walks which start from any two given fixed points have infinitelymany common points with probability 1; whereas for d > 5, two independentrandom walks meet only finitely often, with probability 1.

SELFCROSSING 215

Remark 2. Theorem 21.3 tells us that the paths of two independent random

walks in Zd (d < 4) cross each other. It does not mean that the particles meet

each other.

One can also investigate the selfcrossing of a <i-dimensional Wiener process.

Dvoretzky - Erdos - Kakutani A950) proved the following beautiful theorem:

THEOREM 21.4 For d < 3 almost all paths of a Wiener process have double

points (in fact they have infinitely many double points), i.e. there exist r.v.'s

0 4 almost all paths of a Wiener

process have no double points.

Remark 3. Comparing Theorems 21.1 and 21.4 in the case d = 4 we obtain that

for infinitely many n the paths {So, • • •> $*.} and {^n, 52n+i, • • •} have a point in

common, while for any t > 0 the paths {W(s);0 < s < t} and {W(s);2t <

s < oo} have no points in common with probability 1. This surprising fact is

explained by Erdos and Taylor A961) as follows: for d = 4 with probability 1

the paths {W^(s);0 < s < t) and (W^s); 2t < s < oo} approach arbitrarily close

to each other for arbitrarily large values of t. Thus they have infinitely many

near misses, but fail to intersect. This explanation suggests the following:

Question 1. Characterize the set of those functions /(•) for which

)||=O a.s. (d = 4).2l<u

Chapter 22

Large covered balls

22.1 Completely covered discs in Z2

We say that the disc

Q{r) = {x e Z\\\x\\ < r)

is covered by the random walk {Sk} in time n if

?(x, n) > 0 for every x G Q{r).

Let R{n) be the largest integer for which Q(R(n)) is covered in time n. The

Recurrence Theorem of Chapter 16 implies that

lim R(n) = oo a.s. B2.1)

We are interested in the rate of convergence in B2.1). We prove

THEOREM 22.1 (Erdos - Revesz, 1988, Revesz, 1989/A-B, Auer, 1990). For

any s > 0 and C > 0 we have

exp B(log nI/2 log3 n) 6 UUC(i2(n)), B2.2)

exp (^^(lognloganI/2) 6 ULC(R(n)), B2.3)

exp (C(lognI/2) e LUC(i2(n)), B2.4)

exp ((log nI/2(log log n)�/2"') € LLC(i2(n)). B2.5)

About the limit distribution of R(n) we prove

217

218 CHAPTER 22

THEOREM 22.2 (Revesz, 1989/A, 1989/B). For any z > 0

< limsupPv » v "

> z \ < exp — . B2.6)~

n-oo I logM J\ 4 I

This theorem suggests the following:

Conjecture 1. There exists a A > 0 such that

>,}«p(logn J

Note that we cannot prove even the existence of the limit distribution.

At first we introduce a few notations and prove some lemmas.

Let a(r) be the probability that the random walk {Sn} hits the circle of radius

r before returning to the point 0 = @,0), i.e.

a(r) = P{inf{n : ||5n|| > r} < inf{n : n > l,Sn = 0}}.

Further, let

p@ ~> x) = P{inf{n : n > l,Sn = 0} > inf{n : n > l,Sn = x}}= P{{5rn} reaches x before returning to 0}.

LEMMA 22.1

J|im a(r) log r = tt/2. B2.7)


{inf{n : ||5n|| > r} > inf{n : n > l,Sn = 0}}

C {€@, r2 log2 r) > 0} + ( max ||5*|| < r) .

^0<Jk<r3log''r J

Since by Lemma 19.1

P{f@,r2log2r) =0}«7r/2logr

and by a trivial calculation (cf. Theorem 17.5)

P{ max ||Sfc||<r}=-o(l/logr),0<Jfc<r3log3r

LARGE COVERED BALLS 219

we have

«(') >2 log r

Observe also

a(r) <P{ max ||54|| > r} + PU^logr)�) = 0}.0<Jk<r3(logr)-1

Since by Theorem 17.5

P{ max ||5*|| >r} = exp(-O(logr)),CKJKr^logr)-1

applying again Lemma 19.1 we obtain B2.7).

Remark 1. Lemma 22.1 is closely related to Lemma 17.1.

LEMMA 22.2 There exists a positive constant C such that

Q

log||x||

for any x € Z2 with \\x\\ > 2. Further,

— < liminf p@ ^ x) log ||x|| < limsupp@ /v> x) log ||x|| < -.

12 ||x|| — oo ||x||—oo 2

Proof. Let x = ||x||e'*\ Then by Lemma 22.1 the probability that the particlecrosses the arc ||x||e'* ( 0 if ||x|| is big enough). Since startingfrom any point of the arc ||x||e'* (<?> — tt/3 < xjj < (p + tt/3) the probability that

the particle hits x before 0 is larger than 1/2 we obtain the lower estimate of the

liminf. The upper estimate is a trivial consequence of Lemma 22.1.

Spitzer A964) obtained the exact order of p@ /v> x). He proved

LEMMA 22.3 (Spitzer, 1964, pp. 117, 124 and 125).

7T + OA) ..

p@ ~x) =

4lo7Has l|x|l^°°-

LEMMA 22.4 Let

Yi = t{x\pi)-t{x,pi-l)-l (t = l,2,...)

220 CHAPTER 22

and Zi = —Yi. Then there exists a positive constant C* .such that for any 6 > 0

and f(n) f oo for which n/ f(n) —> oo we have

(c21/2 / \ !/2\

—^~ + c vTRJ B2*8)

l/2>

i + Z2 + ¦ ¦ ¦ + Zn > Son3'4} < exp [ -^- + C*[Tf^) } B2.9)

where

o* = -EY* = 2p{Q^x)

^

( n^ \Ikll < exP ( 77TT) • B2-10)

B2.8) and B2.9) combined imply

. B2.11)

Proof. By a simple combinatorial argument (cf. Theorem 9.7) we get

= k} = (l-p@-x))fc(p@- x)J (A: = 0,1,2,...),= 0,

and

/xm = EV- = (-1)", + f; kmp2qkJk=O

2qk< (-l)mq + Ep2qkk{k + l)--'{k + m-l) = {-l)mqJk=i

where p = p@ /\> x) and ^ = 1 —

p. Hence

|pm| <q + m\pl-mq (m

Similarly

Kl = \EZ?\ < q + mlpl-mq (m = 3,4,.. .)•

Let

2

g(z) = EexpBy1) = qe~z +P

+z,

tp(z) = \og g{z)

and

s = sn= 6o-ln-ll\

Then by Lemma 22.2 and condition B2.10) we have

2

if n is big enough. Hence

(k = 3,4,...) and

\*{s)\ < log2 2 oo Je oo

f fJk=3

K-Jk=3

3 . MM ^s3V+M - 2 <—- + 3< log I 1 H h qs e + pq I - I 2

2' *~

" ' "

Vp7 "/"

2" " "

P2

Similarly, r* M I ,/ M

<^2S2 3 S2S3

llogEexp^ZJl = \v{-s)\ < —— + s e + —-.

L p

Let

Fn(k)=-P{Y1 + Y2 +¦¦¦ + ?„ = k}and define a sequence l/j, U2, • • • of i.i.d.r.v.'s with

pm _ ^.\ — e~^a)gak'p{Yi = k\.

Then

222 CHAPTER 22

where

Hence

Yn > Son3"}

< exp(nip(s) - s6an3/A)2s3'

< exp n

and we have B2.8).Note that the proof of B2.9) is going on the same line but instead of the

sequence {Un} we have to use the sequence {U^} defined by

; = k} = e-*(-V*P(Z! = k)

and we have the lemma.

Now we turn to the proof of B2.5). In fact we prove the much stronger

THEOREM 22.3 (Auer, 1990). For any e > 0 we have

lim sup li,, -II =0 a.s. B2.12)—

|,||<r.'^0 n)

where

rn = exp

Remark 2. Note that B2.5) claims that the disc around the origin of radius

rn is covered in time n. The meaning of B2.12) is that the very same disc

is "homogeneously" covered, i.e. every point of this disc will be visited about

f@, n) ~ log n times during the first n steps. For the one-dimensional analogueof this theorem, cf. Theorem 11.9.

Proof of Theorem 22.3. Clearly

i{x,Pn)i =

y1 + y2 + --- + yw

and by Lemma 22.4 we have

P { sup

•

supll*ll<An

sup n\ ><5<rn3/4}

n*

exp-

2 +C\f(n),where An = exp(n1/2//(n))- Let f(n)lemma we obtain

= (lo8n)e then by the Borel " Cantelli

sup -l|=0 a.S.

which, in turn, implies

lim sup supn~>0°

||*||<An Pn<m<Pn+l

= 0 a.s.

Then replacing n by ^@, N) and recalling that

for all but finitely many N (cf. Theorem 19.4) we obtain the theorem.

In order to prove B2.2) of Theorem 22.1 we prove two lemmas and introduce

a few notations. Let

J'nJ-\0 if Z(x,n)=0,mk = mk{xu x2,.. •, xk; n) = E{I{xu n)I{x2, n)..., I{xk, n))

= P{e(x!,n) >0,e(x2,n) >0,...,e(xfc,n) > 0},

^B) = ux = min{A;, k > 0, Sk = 2},

t=i

Then we have

224 CHAPTER 22

LEMMA 22.5 For any 0 < q < 1 k = 2,3,... we have

mi(u;n-qn)[M{qn) + A - k)mk{xu x2,... ,xk; qn)]. mi(v,n)M(n)

":"'(t)()

where u € Z2 and v 6 Z2 are defined by

mi(v,n) = max mi(x,,n) and mi(u,n) = min mi(x,,n).

In order to present the proof in an intelligible form we prove Lemma 22.5 first

in the case k = 2. That is, we prove

LEMMA 22.6 For any 0 < q < 1 we have

xiqn) +m1(y;qn) - m2{x,y;qn)\

n) <.

l + mi(x-y;n)

Proof.

m2{x,y;n) = P{/(x,n) = 1, J(y,n) = 1}

Cx.n) = l,/(y,n) = 1 | ux = k < uv}Y{ux = k < uv}Jk=O

n

J^ P{/(x, n) = 1, J(y, n) = 1 | uy = k < ux}V{uy = k < ux}

.n) = 1 I ux = k < uy}Y{ux = k<uv}

k=0

Jk=O

Jk=O

n

= ]P P{^(y —

x,n— k) =

k=o

n

+ J^I>{I{x-y,n-k) =

k=o

Consequently we have

(x -y,n) = 1}P |f:(K = *<i/,) + (i/,=*< i/,))}U=o J

= P{/(x -

y, n) = 1}P{/(*, n) = 1 or I(y, n) = 1}= m^x- y;n)[mi{x;n) +m1(y,n) - m2(x, y,n)}

which implies the upper part of Lemma 22.6.

We also have

™2{x,y;n) > ?P{/(y -

x,n- k) = l}P{i/x = * < uy}

qn

T.Jk=O

+ 5Z p{^(x ~

j/' n ~ ^) =

Jk=O

> P{/(x -

y, n - gn) = l}P{/(x, qn) = 1 or J(y, gn) = 1}= m^x- y;n-qn)[mx{x;qn) +mi{y;qn) - m2(x,y;qn)}.


Proof of Lemma 22.5. Let Pk resp. Pjk(r) be the set of permutations of the inte-

integers 1,2,..., k resp. 1,2,... ,r-l,r+l,..., k. Further, let A = A(ii,i2,...,ik;j) =

{u{Xil) < i/{Xii) < ..., u{xik_l) =j< v{xik)}- Then we have

mk{xi,x2,...,xk;n) = ^ P{/(x!,n) =...

= I(xk,n) =

Consequently

= m!(v,n)[.M(n) + A

which implies the upper part of the inequality of Lemma 22.5.

We also have

(U •¦

<

226 CHAPTER 22

r=l

= P{/(u,n-gn) =

xP

= m^u.n- gn)[.M(gn) + (l - k)mk\.Hence we have Lemma 22.5.

Let N = N(n) / oo and k = k(n) / oo be sequences of positive integerswith N(n) < n1/3. Assume that there exists an e > 0 for which k(n) < (N(n))e.Then for any n = 1,2,... there exists a sequence X\ = Zi(n), x2 = x2(n),..., x^ =

xk{n) € Z2{k = k(n)) such that

N -1 < \\xi\\ <N (i = l,2,...,Jk),Nl~e < \\x{ - Xj\\ < N A < i < j < k).

Now we formulate our

LEMMA 22.7 For the above defined X\, x2,... xk we have

(^^( (^^jj). B2.13)

Further, if

N{n) = exp((log nI/2 log3 n) and k{n) = exp((log nI/2) B2.14)then

i, x2,..., xk; n) < exp(-B - 4e) log3 n). B2.15)

Proof. By Lemma 22.5 and Theorem 19.3 we have

logn

Hence we have B2.13). If B2.14) holds then

| 0(logn I I logn

< exp I-B - 4g)lo n

flog") ' 1 = «P("B - 4e) log3

and we obtain B2.15).Now we can present the

Proof of B2.2). Let N{n) be defined by B2.14) and let

ni = (exp(eJ)],

N{n) =expB(lognI/Mog8n),^•+1=exp((logni+1I/2).

Then

where x,- = x,(ni+1) (i = 1,2,..., kj+l). Hence by the Borel - Cantelli lemma

a.s.

for all but finitely many j. Let n, < n < nJ+1. Then

R(n) < R{nj+1) < N{n3) < N(n)

which proves B2.2).Since B2.4) is a trivial consequence of the upper inequality of B2.6) we prove

the upper part of B2.6). In fact we prove a bit more:

THEOREM 22.4 (Revesz, 1989/A). For any z > 0 and for any n = 2,3,...we have

y B2.16)4/

In order to prove Theorem 22.4 let

M = M{n) = exp(C(log n)l'2) (C > 0)

228 CHAPTER 22

and

K = K{n) = <T exp (^-(lognI/2) (C* > 0).

Then for any n = 1,2,... there exists a sequence yx = j/i(n), y2 = 5/2A),..., yK

I/if(n) such that

M-\< \\vi\\ <M (i = l,2,...,K),

MZ'A < \\Vi - yy|| < M A < 1 < j < K).

if C* is small enough. Now we formulate our

LEMMA 22.8

mK(yi,y2,...,yK;n) < exp I——I .

Proof. In the same way as we proved Lemma 22.7 we obtain

«-(-?Hence we have Lemma 22.8.

Proof of Theorem 22.4. Clearly

P{R{n) > exp^QognI/2)} < mK < exp I-—1

which proves Theorem 22.4.

Instead of proving B2.3) we prove the following stronger

THEOREM 22.5 For any 0 < 0 < (tt/ 120I/2,9/10 < <52 < 1 and e > 0 we

have

inf ?(x, n) > -—pr^-(lognlog3 nI/2 t.o. a.s.

I<M)V ~

V^

where

@A — e)—7=v71"

In order to prove Theorem 22.5 let pi@ ^ x),p2@ ^ x),... resp. pi(x /v>

0),p2(x ^* 0),... be the first, second, ... waiting times to reach x from 0, resp.

to reach 0 from x, i.e.

Pi@ '\> x) = inf{n : n > 1, Sn = x},

Pi(x '\> 0) = inf{n : n > Pi@ ^ x), Sn = 0} - pi@ "^ x),

p2@ '\> x) = inf{n : n > />i@ ^ x) + ^(x ^ 0), 5n = x}

p2(x '\> 0) = inf{n : n > p^O '\> x) + Pi(x '\> 0) + p2@ '\> x), 5n = 0}- (Pi@ ~> x) + pi(x ~f 0) + p2{0 ~f x)),...

Let r@ /v> x, n) be the number of 0 /v> x excursions completed before n, i.e.

f 1r@ '\> x, n) = max <» : ^!(Pj@ /v> x) + py(x 'v> 0)) + p,-@ 'v> x) < n > .

I ;=i J

In the proof of Theorem 22.5 the following lemma will be used.

LEMMA 22.9 For any 0 < 0 < (tt/120I/2,9/10 < 62 < 1 and n big enoughwe have

Pin�/2 inf r@^x,,n)<l-<5}<exp(-^^V B2.17)||i||<e«vs \ 60 0 /

(For pn, see Notation 6.)

Proof. Let q=l — p=l— p@ /v> x). Then applying Bernstein inequality(Theorem 2.3) with e = 6p and Lemma 22.2 we obtain

p |r((Wn

n

60log||x|

provided that ||x|| is big enough.Hence

inf n-^2T@^x,pn)<l-6}<-pl infr(° ^ ^^)

^ 1 _

Anp

60 0B2.18)

230 CHAPTER 22

which implies B2.17).

Proof of Theorem 22.5. B2.17) clearly implies that

limmfn�/2 inf r@ ~> x,pn) > 1 - 6

for any 0 < 0 < (tt/120I/2 and 9/10 < 62 < 1.

Observe that Theorem 19.5 and B2.18) imply

inf rfcWs.expf*1/^™)) >1-S i.o. a.s.,

i.e.

inf r@ 'v> x,n) > —p^(lognlogonI/2 i.o. a.s.

||||<M)V ~

^T

which in turn implies Theorem 22.5.

Now, we have to prove the lower inequality of B2.6). In fact instead of provingthe lower part of B2.6) we prove the following stronger

THEOREM 22.6 For any e > 0 and z > 0 there exists a positive integerNo = N0(e,z) such that

P{ inf r@^x,n)>(l-<5)BlognI/2}>exp(-7T2)-e B2.19)||i||<exp(e(zlogn)l/2)

if n > N0,0 < 0 < (tt/120I/2 and 9/10 < <52 < 1.

Proof. Theorem 19.2 and B2.18) imply that for any e > 0 and z > 0 there

exists a positive integer No = No(e,z) such that

P{n~1/2 inf r@ ^ x, pn) < 1 - <$} < e

||i||<e«v^

and

P \pn < exp l-J I > exp(-7rz) - e

if n > iV0. Consequently

p( inf rfo-^x^xpf-)) ^(l-^n1/2}

( inf rfo^x^xpf-)) > (l-<5)n1/2,pn<exp(-)le^ \ \Z) ) Z JZ J

> Pi inf r{0^x,pn) >(l-<5)n1/2,pn<exp(-)|> P{ inf T@^x,pn)>(l-6)nl'2}-I>{pn>exp(-)}

||x||<e®v^ I \zJ J

> 1 — e — A — exp(—7T2)) — e = exp(-Kz) — 2e


if n > No.Hence we have Theorem 22.6 as well as Theorems 22.1 and 22.2.

Theorem 22.3 clearly implies that for any fixed x € Z2

lim B2-20)

It is worthwhile to note that for fixed x in B2.20) a rate of convergence can

also be obtained. In fact we have

THEOREM 22.7

where a(x) is a positive constant depending only on x.

Remark 3. Theorem 22.7 and Theorem 19.4 combined imply

lim( log n

n-°° \(loglognI+ey ?@, n)= 0 a.s.

Before presenting the proof of Theorem 22.7 introduce the following notations:

let Hi, H2,... be the local time of 0 during the first, second, ... 0 '\> x excursions,i.e.

= 52 = x) + pi{x ~e 0) Si,

where

i?3 = ^@ ^ x) + pi{x ~c 0) + p2@ ~c x) + p2(x ~c 0) + p3{0 ~c x).

Observe that H^ H2,... are i.i.d.r.v.'s with distribution

P(S1 = *)=P(?@,Pi((W *))=*)

= A - p@ -^ ijj^-^O -^ x) (* = 1,2,...).

Consequently

S! - (p@ - x))�J = (p@ - p@ - x))


232 CHAPTER 22

and

5,'n^ BnloglognI/2

-

p((W z)

where

z) + pi(z ^ 0)),x)+ Pi{x ^ 0) + p2@ ~> x) + p2(z ~> 0)) -

Since

S1 + s2 + ... + ST((K.x,n) < ?@, n) < Ex + 52 + ¦ ¦ ¦ + ST@^

and the sequence r@ /v> x, n) takes every positive integer we have

imSUP7;r;. . , ,,= 2 ;r

n-oo Br@ ^ z,n) log log r@ ^ XjU)I/2 p@ ^ x)

By the law of large numbers

a.S.

n—oo

Hence

1T-.S<SPB?@,n)loglog{@,n))'A=( p@ -«) j ='«*' «*

and we have Theorem 22.7.

Remark 4. Theorem 11.26 claimed that the favourite values of a random

walk in Z1 converge to infinity. It is natural to ask the analogue question in

higher dimension. In the case d > 3 the Polya Reccurence Theorem implies that

the favourite values are also going to infinity. Comparing Theorem 19.4 and

Theorem 19.7 we find that in Z2 the favourite values are also going to infinity.Theorem 22.3 also says that the rate of convergence is not very slow. In fact we

get

THEOREM 22.8 Let d = 2 and consider a sequence {xn} for which ?(xn, n) =

f (n). Then for any e > 0 we have

liminr rj- >1/,/1—: N .,, .= oo a.s.

"-<» exp((lognI/2(loglogn)-1/2-«)

Remark 5. Replacing the random walk in Theorems 22.1 and 22.2 by a Wiener

sausage Br(T) one can ask whether these theorems remain valid. A harder

question is to study the case where r = rT | 0. In fact Theorem 17.4 impliesthat if rT < T~^ogT>>' (with some e > 0) then the analogues of Theorems 22.1

and 22.2 cannot be true anymore.

22.2 Discs covered with positive density

The results of Section 22.1 claimed that the radius of the largest covered disc

is about exp((lognI/2). Now we are interested in the relative frequency of the

visited points in a larger disc.

In order to formulate our results, introduce the following notations:

I(xn)-il if ^'n)>0''^-[O if e(*,n)=0,

K(N,n) = (N2*)-1 ? I(x,n);zeq(N)

i.e. K(N,n) is the density (relative frequency) of the points of Q{N) covered bythe random walk {Sk,0 < k < n).

Our first theorem claims that if we consider the disc of radius exp((logn)a)(a < 1) or even of radius exp(logn(loglogn)~2~e) (e > 0) then the density of the

covered points converges to one a.s. In fact we have

THEOREM 22.9 (Auer - Revesz, 1989). For any e > 0

=l a.s.

Proof. Consider

where

N = Nn = exp

Then by Lemma 22.2 we have

234 CHAPTER 22

provided that ||x|| < N. Hence

± E(l-/(x,pn))<exp(-C(lognI+e),

and by the Markov inequality for any 6 > 0

P{1 - K(N,pn) >S}< S-'

which, in turn, by Borel - Cantelli lemma implies that

Jim A - K{N,pn)) = 0 a.s. B2.21)n—»oo

Let m = mn = [exp(n(lognI+e)]. Then by Theorem 19.5 mn > pn a.s. for

all but finitely many n. Hence B2.21) implies

\im(l - K(N,m)) =0 a.s.

Observe that given the choice of m and N we have

\ogm \N

( log m

J 6XP

and we obtain

f-—(

limKlexpf-— TiT7)'m)=1 a-s-n^°°

\ \(loglogmJ+«/ /

Consequently we also have

1- TS I ( l°g"Wl \ \ ,

hm K exp -—: rz-r- , mn =1 a.s.»»« v vA°g1°gm) / /

This proves the theorem.

Theorem 22.9 tells us that almost all points of the disc Q(exp(log n)a)(l/2 <

a < 1) will be visited by the random walk {So, Si,..., Sn}. At the same time byTheorem 22.1 we know that some points of Q(exp((logn)a)) will surely not be

visited. We can ask how many points of Q(exp((logn)a)) will not be visited, i.e.

what is the rate of convergence in Theorem 22.9? However, it is more interestingto investigate the geometrical properties of the non-visited points. For example:what is the area of the largest non-visited disc within Q(exp((logn)a))? Bynon-visited disc we mean a disc having only non-visited points. The followingtheorem claims that with probability 1 there exists a non-visited disc of radius

exp((logn)^) within the disc Q(exp((logn)a)) for every /3 < a provided that

a > 1/2.Let

Q(u,r)={x€Z2,\\x-u\\<r}.

Then we have

THEOREM 22.10 (Auer - Revesz, 1989). Let

1/2 < a < 1 and 0 < a.

Then there exists a sequence of random vectors Ui, u-i,... such that

Q(un,exp((logn)^)) C Q@,exp((logn)a)) = Q(exp((logn)a))

and

I{x,n)=0 for all x € Q(un,exp((logn)^)).

In order to prove Theorem 22.10, first we introduce a notation and present two

lemmas.

Let N > 0 and ux, u2,..., uk € Z2(k = 1,2,...) be such points for which the

discs

A = 1,2,...,*)

are disjoint. Denote by

mk{QuQ2,...,Qk-,n) = P{Vi = 1,2,...,/:, 3y, € Q, such that /(y,,n) = 1}

the probability that the discs Q\,Qi,---,Qk are visited during the first n steps.

LEMMA 22.10 Let

exp((logn)a) < ||rx|| < n1/3 N <e^

and

O<0 < a< 1.

Then

for a suitable constant C > 0.

236 CHAPTER 22

Proof. It is easy to see that

P{/(u,2n) = l}>m1(Q(u,iV);n) min P{?(u,2n) - f(u,n) > 0 | Sn = y}.y€Q{uN)

Hence by Theorem 19.3

mi(Q(u,N);n) < )r^-{ < 1 - C(logn)-'.

fe)Hence Lemma 22.10 is proved.

LEMMA 22.11 Let

(i) 0 < 0 < a < 1,

(ii) iV=exp((lognn,(iii) ui, u2,..., ujt € Z2 be a sequence for which

\\ui\\<n1"- A = 1,2,...,*),

||u, - tty|| > exp((logn)a+e) A 0 where Q, = Q(ui,N) (i = 1,2,..., k).

Proof. Clearly

= P-fujL^VQy are visited before n and Qt- is the last visited disc}}< P{U*=1{the discs Qi,..., Q,-i, Qi+i, ¦ -iQk are visited before n}}x maxmaxPlQ. is visited before 2n\ Sn = x).

ijtj xeQ;

Hence by Lemma 22.10

mk{Qi,...,Qk;n)

< (ibmi<-i{Qi> ¦ ¦ -'Qi-uQi+u ¦ ¦ -,Qk;n) - {k - l)mfc(Ql5... ,Qk;n)

x(l-C(logn)a-1)

and

mk{Qi,...,Qk;n)¦¦¦,Qk]n)

Since1-a

for any 0 < a < 1 and Jk>lwe have

mk{Q1,...,Qk;n)

by induction

mk{Q1,...,Qk;n) <

exp (-dlogn)*-1^) <

V t=2 */

and we have Lemma 22.11.

Remark 1. Lemma 22.11 is a natural analogue of Lemmas 22.5 and 22.7.

Proof of Theorem 22.10. Let

1/2 < a< a + e < 1, 0 < a

and

iV = exp((logn)/?).Then there exist k = k(n) = exp((log n)a) points Ui, u^,..., uk such that

||u,||<exp((logn)a+e) (i= 1,2,...,*),

and

\\xi - xy|| > exp((logn)a+e/2) (i,i = 1,2,..., k; i / j).

Then by Lemma 22.11

mk{QuQ2,...,Qk;n) < exp(-C(lognJ"�)

238 CHAPTER 22

where Qi = Q(ui,N). Choosing n = n, = e3 the Borel - Cantelli lemma impliesthat with probability 1 at least one Q, (i = 1,2,..., k(rtj)) is not visited till nifor all but finitely many j.

Let rij < n < rij+i. Then with probability 1 there exists a u with ||uj| <

exp((> + l)a+e) such that the disc Q{u,N) {N = Nj = exp^logn,-)")) is

not visited before n if j is large enough. Consequently for all but finitelymany n there exists a u0 = uo(n,uj) € Q(exp(logn)a+2e) such that Q(uo,N) C

Q(exp(log n)a+3e) is not visited before n. This proves Theorem 22.10.

Now we consider the density K(N,n) for even larger N. The case N = na

will be investigated and for any 0 < a < 1/2 we prove that K(na,n) has a limit

distribution. In fact we have

THEOREM 22.11 (Revesz, 1989/A). For any 0 < a < 1/2 there exists a

distribution function Ga{x) with Ga@) = 0,Ga(l + 0) = 1 and

limP{K([na],n) < x) = Ga{x) (-00 < x < oo)

for almost all x.

At first we present a few lemmas.

LEMMA 22.12 Let na(logn)-^ < ||x|| < Cna @ < a < 1/2,0 > 0,C > 1).Then

Proof. It is a trivial consequence of Theorem 19.3.

LEMMA 22.13 Let x and y be two points of Z2 such that

na(logn)^ < ||x||, ||y||, ||x - y|| < Cna @ < a < 1/2,0 > 0,C > l).

Then

lim m2{x, y; n) =A 2a)

. B2.22)n—»oo 1 — Q;

Proof. Lemmas 22.6 and 22.12 imply

-

y\ njlm^x; n) + m^y; n)]<

x- y;n)

B2.23)= + o f^) .

1 - a V log n J

Similarly

m2{x,y;n) >

mi(x- y\n-qn)[mi(x;qn) + m1(y;gn) - m2{x,y;qn)\

Consequently

H^^). B2.20

B2.23) and B2.24) together imply B2.22).The next lemma is an extension of Lemma 22.13.

LEMMA 22.14 . Let xu x2,..., xk (k = 1,2,...) be a sequence in Z2 such that

na(logn)~^ < \\xi - Xj\\, \\xi\\ < Cna

where 0 < a < 1/2,/? > 0, C > 1,1 < i < j < k. Then

Proof. Lemmas 22.5 and 22.12 imply

O (mk < -.

V

(f1Og"

By induction we obtain

240 CHAPTER 22

Similarly

rrik > —

j-

Consequently

and by induction we obtain Lemma 22.14.

Proof of Theorem 22.11. Let A(s, n) be the set of all possible s-tuples(xi, x2,...,xs) of Q(na) with the property

na„-.„,„-. -,„_

logn

Then

(X!,«a x.)€A(t,n)

and by Lemma 22.14 we obtain

Urn E((tf(n",n)n = A - 2a)']& (l - (l - j) 2a

Consequently we have Theorem 22.11 with a distribution Ga(-) satisfying

(z) = A - 2o)' ) 2]lY1

Remark 2. The above equality easily implies that Ga(l — e) < 1 for any0 < a < 1/2 and e > 0 and in turn limsupn_>00 K(na,n) = 1 a.s. for any0 < a < 1/2.

It is natural to ask the following

Question. Is it true for any 0 < a < 1/2 and e > 0 that

Ga(e) > 0?

22.3 Completely covered balls in Zd

Theorem 22.1 describes the area of the largest disc around the origin covered

by the random walk {Sk,k < n}. In Zd (d > 3) the analogous problem is

clearly meaningless since the largest covered ball around the origin is finite with

probability 1. However, one can investigate in any dimension the radius of the

largest ball (not surely around the origin) covered by the random walk in time

n. Formally speaking let

Q{u,N) = {x:xeZd, \\x - u\\ < N}

and R*(n) = R*(n,d) be the largest integer for which there exists a random

vector u = u(n) G Zd such that Q(u,R*(n)) is covered by the random walk in

time n, i.e.

?(x, n) > 1 for any x G Q{u, R*{n)).

Then we formulate our

THEOREM 22.12 (Revesz, 1989/C). For any e > 0 and d> 3 we have

(logn)^+e eUUC{R*{n)) B2.26)

ande

e LLC(R*(n)). B2.27)

At first we present a few lemmas.

LEMMA 22.15 For any d>3 there exists a positive constant C& such that

P{Sn = x for some n} = P{J{x) = 1} = ^J^ {R - oo)

where R = \\x\\ and

jix\ =i° *

' | 1 o

?(*,n) =0/or every n = 0,1,2,...,otherwise.

242 CHAPTER 22

Remark 1. A somewhat weaker version of Lemma 22.15 was found by Erdos -

Taylor A960/A).It is worthwhile to mention the following analogue of the above lemma for

Wiener process.

LEMMA 22.16 (Knight, 1981, p. 103). For any d>Z

( r \ d~2

P{W(t) e Q{u,r) for some t} = I —

J

provided \\u\\ = R > r.

Put a = r, b = R, c = oo, then this lemma is a simple consequence of Lemma

17.1.

Proof of Lemma 22.15. Clearly

P{Sn = x} = f^P{Sk = x,S^x,j = 0,1,2,...,k- n_fc = 0}

and

n=0 fc=0n=0

n=0

Since by Lemma 16.5

,y = 0,1,2,... ,fc - l}P{5n_fc = 0}

= x,S,?x,j = 0,1,2,...,k-l}.k=0

? P{Sn = x} = (Kd + o(l))R2-d (R - oo)n=0

B2.28)

and by Lemma 16.2

we obtain

f;P{5n=0}<oo,n=0

P{J(x) = 1} = ?p{5fc = x,Sj ? x,j = 0,1,2,...,k - 1} =

k=0

Hence we have the Lemma.

o{l))R2-d.

Remark 1. For a more exact form of B2.28) see Spitzer A964), Pi (p. 308)and Problem 5 (p. 339).

LEMMA 22.17 For any 0 < a < 1 and L > 0 t/iere eztsts a sequence xi, x2,...,

Xy o/ t/ie points of Zd such that

L< \\xi\\ < L + l (i = l,2,...,T),

||xt-x;|| >La {i,j = 1,2,...,T;i^j),

where K = K(d) is a positive constant depending on d only.

Proof is trivial.

LEMMA 22.18 Let

Dk+l-Dka = (Xi = (Xi(k) =

where

and define T = Tk and x\,X2,...,xt as in Lemma 22.17. Then for any L bigenough we have

P{Q@, L) is covered eventually}

i, X2,.. •, xj are covered eventually} < e~^T~x\

Proof. Define the sequence au aj,...,^ (k = 1,2,...) by

Dk+i_Dk+i-i°* =

Dk+i _ x(t = 2,3,...,fc).

Assuming that

Dk _ Dk-i Dk -I0 < e < —rr-r: <

we have

0 < c^ < a2 < ... < ak < 1 (k = 1,2,...)and

(o,-+1 - ai){d - 1) < Oi{d - 2) (i = 1,2,..., k)where a-k+i = 1-

Let xtl, x,-2,..., x,T be an arbitrary permutation of the sequence x1? x2,..., xy.

Consider the consecutive distances

244 CHAPTER 22

Assume that among these distances /i resp. l-i resp. ... /* are lying between Lai

and La* resp. La2 and L� resp., ..., Lak and 2L.

Then (by Lemma 22.15) the probability that the random walk visits the points

1,-j, Xi2,..., XiT in this given order is less than or equal to

Taking into consideration that the number of those j's (l < j' < T) for whichLai < ||x, — i,|| < Lai+l (where s is a fixed element of the sequence A,2,..., T))is less than or equal to

we get

P{xi, X2,... ,xt are covered eventually }

v A

I

= ELi=l

<

if L is big enough and we have Lemma 22.18.

In the same way as we proved Lemma 22.18 we can prove the following:

LEMMA 22.19 For any L big enough and u 6 Zd we have

P{Q(u,L) is covered eventually} < C*Lde-{T-l)

where

D* -1 _ d-l

k is an arbitrary positive integer and C* = C*(d) is a positive constant.

Proof of B2.26). Let L = [(logn)'] with

c al I ( Dk-l Yle 6 +{

Then T > (log n)^ with some t/j > I and we obtain our statement B2.26) ob-

observing that

lim 0k = j-t—-7 7-7—-r + e.fc-00

*

(d - 2) - e(d - 1)In order to prove B2.27) we present a few further lemmas.

LEMMA 22.20 There exists a constant K > 0 such that

,n) > 0} > (^) R2~d R=\\x\\

if n > KR2 where C& is the constant of Lemma 22.15.

Proof. Lemma 16.5 easily implies

KR2 ts

n=l

if K is big enough, which, together with the method of proof of Lemma 22.15

implies Lemma 22.20.

Let

and define

T\ = n H

tjji - inf{k : k> TUSn = Sk} - tx,

T2 = Tx+Vl + IOog"K^1!.V»a = inf {fc : fc > t2, Sn = Sk} - r2,...

Clearly, with a positive probability (depending on n) t/^ is not defined. However,we have

LEMMA 22.21 There exists a constant C > 0 such that

P{ max tb{ <

Proof. Clearly for any C > 0 there exists a constant 0 p > 0.

246 CHAPTER 22

Since

\\STl - Sn\\ <

and 0i, 02, • • • are i.i.d.r.v.'s we have

- Yp\ >n""!

for a suitable C > 0. Hence we have Lemma 22.21.

LEMMA 22.22 Let

A = An = {max 0,;< C(logn)^} and B(n) =

Then among the events B(n) only finitely many will not occur with probability 1.

Proof is trivial.

Proof of B2.27). Let x e Zd satisfying the inequality

Then (by Lemma 22.20)

;n)^T])-?(z,n)=0}<l-

Hence the conditional probability (given An) that x is not covered is less than or

equal to

l - ^(log»)<-<'-*>-1><'-'>)L < exp (-^(logn)^�)) .

Consequently the conditional probability that there exists a point x G

Q((\ogn)(d~l) l~e,5n;(i) being not covered is less than or equal to

This fact together with Lemma 22.22 proves B2.27).Theorem 22.12 tells us that the path of the random walk in its first n steps

covers relatively big balls. It is natural to ask where these big covered balls are

located in Zd. For example we might ask about the radius of the largest covered

ball within Q(u,n). In fact let p(n) be the largest integer for which there exists

an r.v. u = u(n) 6 Rd such that ||u|| < n and Q(u,p(n)) is covered by the

random walk eventually, i.e.

J(x) = 1 for any x G Q(u,p(n)).

As a trivial consequence of Theorems 22.12 and 18.5 we obtain

THEOREM 22.13 For any n big enough, d>3 and e > 0 we have

(logn)^T"e < p{n) < (logn)?^+e a.s.

Remark 2. In Section 22.1, for d = 2 we investigated the area of the largestcircle around the origin covered by the random walk {Sk,k < n}. In the presentSection the volume of the largest covered ball in Zd d > 3 (not surely around the

origin) was investigated. Clearly the latter question can be studied in the case

d = 2 too but nothing is known about this problem.

Remark 3. Applying Theorem 7.1 (cf. also the proof of Theorem 19.6) one can

obtain that

Cd{\ogn)l'deLLC{Rd{n)}

for a suitable C^ > 0. It is a somewhat weaker statement than B2.27) but in

some sense it says a bit more than B2.27). In fact we can obtain (by Theorem

7.1) that:

for any n big enough there exists a positive random integer un < n — Kd log n

such that the path {SVn, SUn+l,..., Sl/n+Kd\ogn} covers a ball of volume Cd(\og nI/*.Remark 3 suggests to ask about the connection between the location of the

favourite value and the largest covered ball. For example, we can propose the

following

Question. Let Q(u(n),R*(n)) be the largest covered ball and let xn G Zd be a

favourite value, i.e ?(xn,n) — f(n). Is it true that

xn e Q{u{n), R*{n)) i.o. a.s.?

248 CHAPTER 22

22.4 Once more on Z2

The results of Section 22.3 suggest the question whether the largest completelycovered disc is located around the origin or there exists a larger completelycovered disc located somewhere else. The answer to this question is unknown.

However, we present the following:


lim sup J2 I{x,n) = l,

where

^'^"{o if ?{x,n)=0.

Proof. By Remark 2 of Section 22.2, for any e > 0 and 6 > 0 there exists a

0 < 7 = "i(e,6) < 1 such that

>-7 (n = l,2,...). B2.29)

Consider the points n0 = 0, nx = [c^\ > n2 =

- By B2.29)

where, f 1 if Sj; = x at least for one nk < j <

k^ ' '

| 0 if 5,-^z for every n* < j' < nk+l.

Then choosing C big enough we obtain the Theorem.

To see the meaning of the above theorem it is worthwhile to compare it with

Theorems 22.9, 22.11 and Remark 2 of Section 22.2.

Chapter 23

Speed of escape

Theorem 18.5 on the rate of escape suggests that the sphere {x : ||z|| = R} is

crossed about R times by the random walk if R is big enough and d > 3. In fact

if ||5n|| = y/n for every n then HxeZ(R) f(z,°°) = O(R) where

Z{R) = {x : x e Zd,\\\x\\ - R\ < l}.

Introduce also the following notations:

j(\-f° if ?(*,") =0 for every n = 0,1,2,...,* '

[1 otherwise

and

z€Z{R)

i.e. J(x) = 1 if x G Zd is visited by the random walk and 0(R) is the number

of points of Z(R) visited eventually by the random walk. On the behaviour of

0(R) I have the following:

Conjecture 1. For any d > 3 there exists a distribution function H(x) = Hd{x)for which H@) =0 and

lim P (^ < x\ = H{x) (-00 < x < oo).—»°o

Unfortunately I cannot settle this conjecture but the analogous question for

a Wiener process can be solved. In order to present the corresponding theorem

we introduce the following notations.

Let W(t) = {W^t), W2{t),... ,Wd{t)} {d > 3) be a Wiener process.

249

250 CHAPTER 23

Definition. W(t) is crossing the sphere {x : \\x\\ = R} 6 = 0(R) times if 6(R) is

the largest integer for which there exists a random sequence 0 < o^ = ax (R) <

ft = ft(JK) < a2 = a2{R) < ft = 02{R)... < a9 = ae{R) < 0e = 06{R) < oo

such that

\\W(t)\\ <R if t < au

\\W{ai)\\=RtR-l<\\W{t)\\<R+l if ^ <*</?!,

\\W{0l)\\=R-lorR + l,\\W(t)\\^R if ft < t < a,,

\\W{a2)\\ = R, R-l< \\W{t)\\ <R+1 if a2 < t < 02,

\\W{02)\\ =R- lor 12 + 1, \\W(t)\\ ?R if ft<*<as,...

\\W{a,)\\ =R, R-K\\W{t)\\ <R+l if a0<t<00,

\\W@9)\\ =R + l\\W(t)\\>R if t>09.

F(R))~l will be called the speed of escape in R.

THEOREM 23.1 (Revesz, 1989/A).

<•} — •-' «->

The proof is based on the following:

LEMMA 23.1

P{6(R) = k} = A(l - A)*� (A: = 1,2,...)

w/icrc

nand

where B{R,t) = {inf{s : s > t, \\W{s)\\ = R-l} > mf{s : s > t, \\W{s)\\ = R+l}.

Remark 1. Note that the last formula for p(R) comes from Lemma 17.1. ByLemma 22.16

2

= P{\\W(t + s)\\<R ioTsomes>0\\\W(t)\\ = R + l}.

SPEED OF ESCAPE 251


P{6{R) = 1} = A,

P{0(R) = 2}/ / R \d-t\ / R \d~2 (1- (-= )+P(R)[~?—7i PWi1

d-2-\

where q(R) = 1 — p{R)- Similarly

PF(r) = k)

i-a\ *-i

\k-\=AA-A)'


Observe that

IA"

p(R) 1 - A - jL)--*~

p(/2)(rf - 2)

Since p(R) -> 1/2 (i2 -» oo) we have

Lemma 23.1 together with B3.1) easily implies Theorem 23.1.

Studying the properties of the process {6(R),R > 0} the following questionnaturally arises: does a sequence 0 < Ri < i22 < • • • exist for which

lim Rn = oo and 0(i2j = 1 i = 1,2,...?n»oo

The answer to this question is affirmative. In fact we prove a much strongertheorem. In order to formulate this theorem we introduce the following

Definition. Let ijj(R) be the largest integer for which there exists a positiveinteger u = u(R) < R such that

6(k) = 1 for any u < k < u + tp(R).

It is natural to say that the speed of escape in the interval (u,u + i/j(R)) is

maximal.

252 CHAPTER 23

THEOREM 23.2,.

. log log Rtp{R) > t.o. a.s.

log 2

Proof. Let

f[R)-log 2

'

A(R) = {6{k) = 1 for every R < k < R + f{R)}

and

C(R,t) = {\\W(t)\\ = R + f(R) + l}.Then

?{A{R)} = JI P{B(R+j,t) | \\W{t)\\ = R+j);=o

x(l -P{\\W(t + s)\\ < R + f(R) for somes > 0 | C(R,t)})1 d-2

~

logi? 2R

and

logii!log 2

if log log A/log 2 < S = o(R). In the case 5 > O(R) the events A(R) and

+ 5) are asymptotically independent. Hence

and for any e > 0 if n is big enough we have

hS))<(^-=| (loglognJ(l+?)2

iEl 5_r!?t!1L| V /' log 2 I

which implies Theorem 23.2 by Borel - Cantelli lemma.

Conjecture 2.

hm :—7——= a.s.

tf-oo log log it log 2

Theorem 23.2 clearly implies that 0(R) = 1 i.o. a.s. It is natural to ask: how bigcan 6(R) be? An answer to this question is

SPEED OF ESCAPE 253


6{R) < 2(d-2)-1{l+e)R\ogR a.s.

if R is big enough and

9{R) > A - e)R log log log R i.o. a.s.

Since this result is far from the best possible one and the proof is trivial we

omit it.

Remark 2. Conjecture 1 suggests that 0(R) ~ R. Instead of investigating the

path up to oo consider it only up to px = min{fc : k > 0, Sk = 0}. Takinginto account that P{pi = oo} > 0 if d > 3 we obtain T,xez{R) ?{x,Pi) ~ R

with positive probability. Investigating the case d = 1 by Theorem 9.7 we get

E?«€*(*) Z{x,Pi) = E(?(*!,Pi) + ?(-*!,Pi)) = 2 for any R G Z\ We may ask

about the analogous question in the case d = 2. By Lemma 22.1 we obtain

Lemma 17.1 suggests that the probability of returning to the origin from Z(R-l)before visiting Z(R) is O(R~l(log R)~l). Hence we conjecture that T,xez{R) €{xiPi~ O(R); for example,

= 0(R) as

\x€Z(R) )

for any d > 2.

Chapter 24

A few further problems

24.1 On the Dirichlet problem

Let U be an open, convex domain in R2 which is bounded by a simple closed

curve A. Suppose that a continuous real function / is given on A. Then the

Dirichlet problem requires us to find a function u = u(x, y) which

(i) is continuous on U + A,

(ii) agrees with / on A,

(iii) satisfies the Laplace equation

d*u d2u

dy2_

A probabilistic solution of this problem is the following. Let {W(t),t > 0}be a Wiener process on R2 and for any z ?U define Wz(t) = W{t) + z. Further,let az be the first exit time of Wz(i) from U, i.e.

oz = mm{t : Wz{t) <E A} (zG U).

Then we have

THEOREM 24.1 The function

u(z) = Ef(cz)

is the solution of the Dirichlet problem.

255

256 CHAPTER 24

The proof is very simple and is omitted. The reader can find a very nice

presentation in Lamperti A977), Chapter 9.6.

Here we present a discrete analogue of Theorem 24.1. Instead of an open,

convex domain U we consider a sequence Ur (r = 1,2,...) of domains defined as

follows.

Consider the following sequences of integers

2 < ... < anr,

c2,. ..,&nr_i <

satisfying the conditions

A) 6t+1 < ct, ct+1 > bi (i =1,2,. /., nr- 2),

B) Oi+1-

a,- > or, c,- b{ > ar,

with some a > 0 and nr = 2,3, Now let

Condition A) implies that Ur is connected. Condition B) has only some minor

technical meaning. Let Ar be the boundary of Ur and define a "continuous"

function /,.(•) on the integer grid of Ar, where by continuity we mean:

For any e > 0 there exists a 6 > 0 such that \f{zi) — f(z2)\ < ? if ||zi — z2\\ < 6r

where zuz2 € ArZ2.Now we consider a random walk {Sn;n = 0,1,2,...} on Z2 and for any

ze{Ur + Ar)Z2 we define

SP = Sn + z (n = 0,l,2,...).

Let az be the first exit time of S,W from Ur, i.e.

az = min{n : 5^ G Ar}.

We wish to prove that

«(*) = E/EW)is the solution of the discrete Dirichlet problem, meaning that

(i) u is "continuous" on (Ur + Ar)Z2, i.e. for any e > 0 there exists a 6 > 0

such that if zuz2 € (Ur + Ar)Z2 and ||zi-Z2|| < ^»", then |uBx) -u(z2)\ < ?,

(ii) u agrees with / on ArZ2,

A FEW FURTHER PROBLEMS 257

(iii) u satisfies the Laplace equation, i.e.

(u(x + 1, y) - 2u(x, y) + u{x - 1, y))+ (u{x, y + 1) - 2u(x, y) + u(x, y

- 1)) = 0

whenever (x, y), (x + 1, y), (x, y + 1), (x - 1, y), (x, y- 1) <E (tfr + Ar)Z2.

(ii) is trivial, (iii) follows from the trivial observation that

u(x, y) = -(u(x + 1, y) + u(x - 1, y) + u(x, y + 1) + u(x, y- 1))

4

if (x, y) satisfies the condition of (iii).In order to prove (i) we present a simple

LEMMA 24.1 For any e > 0 there exists a 6 > 0 suc/i t/iat if

zeUrZ\ qeArZ2, \\z-q\\<6r

then

Consequently\Ef(S{M)M)-f(q)\<e\

Proof is simple and is omitted.

In order to prove (i) we have to investigate two cases:

(a) zuz2eUrZ\

(/?) one of zi, z2 is an element of ArZ2 and the other one is an element of UrZ2.

In case (/?) our statements immediately follow from Lemma 23.1. In case (a)assume that oZl < aZ2 and observe that

Since S(*2V*J = S^tM2)^'^(<rs^){OMi)) applying Lemma 23.1 with q = S

and z = S^(aZl) we obtain (i).

Remark 1. Having the above result on the solution of the discrete Dirichlet

problem, one can get a concrete solution by Monte Carlo method. In fact to getthe value of u(-, •) in a point zq = (x0, yo) & Ur observe the random walk starting

258 CHAPTER 24

in zq till the exit time aZn and repeat this experiment n times. Then by the law

of large numbers

lim n'1 ? /(sH(ffJ) = u(x0, y0) a.s. B4.1)»oo

*—¦"n—»oo

where Si,S2>... are independent copies of a random walk. Hence the average

in B4.1) is a good approximation of the discrete Dirichlet problem if n is big

enough. A solution of the continuous Dirichlet problem in some zo or in a few

fix points can be obtained by choosing r big enough and the length of the stepsof the random walk small enough comparing to the underlying domain.

24.2 DLA model

Let A\ C Ai C ... be a sequence of random subsets of Z2 defined as follows:

Ax consists of the origin, i.e. A\ — {0},

A2 = A\ + y2 where y2 is an element of the boundary of At

obtained by the following chance mechanism. A particle is released at oo and

performs a random walk on Z2. Then y2 is the position where the random walk

first hits the boundary of Ax.The boundary of a set A C Z2 is defined as

dA = {y : y E Z2 and y is adjacent to some site in A, but y ? A}.

For example, dA, = {@,1), A,0), (-1,0), @, -1)}.Having defined An,An+i is defined as An+i = An + yn+i where yn+i is the

position where the random walk starting from oo first hits dAn.In the above definition the meaning of "released at oo" is not very clear.

Instead we can say: let

Rn = inf{r : r > 0, An C Q{r) = {x : \\x\\ < r}}.

Then instead of starting from infinity the particle might start its random walk

from (R^,0) (say). It is easy to see that the particle goes round the origin before

it hits An (a.s. for all but finitely many n). This means that the distribution of

the hitting point will be the same as in case of a particle released at oo.

Many papers are devoted to studying this model, called Diffusion Limited

Aggregation (DLA). The reason for the interest in this model can be explained

by the fact that simulations show that it mimicks several physical phenomenawell.

The most interesting concrete problem is to investigate the behaviour of the

"radius"

rn = max{||x|| : x? An}.

Trivially rn > (m/ttI/2 and it is very likely that rn is much bigger than this trivial

lower bound. Only a negative result is known saying that rn is not very big. In

fact we have

THEOREM 24.2 (Kesten, 1987). There exists a constant C > 0 such that

limsup n~2/3rn < C a.s.

n—»oo

The proof of Kesten is based on estimates of the hitting probability of dAn.He proved that there exists a C > 0 such that for any y 6 dAn we have

P{yn+i = y}< Cr-W. B4.2)

In order to get a lower estimate of rn we should get a lower estimate of the

probability in B4.2) at least for some y G dAn. Auer A989) studied the questionof how one can get the lower bounds of the hitting probabilities of some pointsof the boundaries of certain sets (not necessarily formed by a DLA model). He

investigated the following sets:

Bi = {(-r,0), (-r + 1,0),..., (r - 1,0), (r,0)},B2 = Bx + {@, -r), @, -r + 1),..., @, r - 1), @, r)},B3 = {x = (xi, x2) : \xi\ + \x2\ = r}.

Consider the point y = (r,0). Then the probability that the particle coming from

infinity first hits y among the points of dBi(i — 1,2,3) is larger than or equal to

Cr'1'2 if i = 1,2

and

(gr)-1'3 if i =

with some C > 0.

260 CHAPTER 24

24.3 Percolation

Consider Z2 and assume that each bond (edge) is "open" with probability p and

"closed" with probability 1 —

p. All bonds are independent of each other. An

open path is a path on Z2 all of whose edges are open.

One of the main problems of the percolation theory is to find the probability0(p) of the existence of an infinite open path. Kesten A980) proved that

0(P)\>O if p>l/2.

The value 1/2 is called the critical value of the bond percolation in Z2.An analogous problem is the so-called site percolation. In site percolation the

sites of Z2 are independently open with probability p and closed with probabilityq = 1 —

p. Similarly as in the case of the bond percolation a path of Z2 is called

open if all its sites are open and we ask the probability @*(p) of the existence of

an infinite open path. The critical value of the site percolation in Z2 is unknown,but T6th A985) proved

0*(p) = 0 if p < xAQ ~ 0,503478

where x0 is the root lying between 0 and 1 of the polynomial

We call the attention of the reader to the recent survey of Kesten A988) on

percolation theory.

And God said, "Let there be lights in the

firmament of the heavens to separate the

day from the night; and let them be for

signs and for seasons and for days and

years."

The First Book of Moses

III. RANDOM WALK IN RANDOM

ENVIRONMENTNON-ACTIVATEDVERSION

www.avs4you.com

Notations

1. ? = {... ,E_2,E_i,Eo,Ei,E2,...} is a sequence of i.i.d.r.v.'s satisfying0 < Ei <1- 0 with some 0 < 0 < 1/2 called environment.

2.

3.

j^n^T^Pe},^?,?}; see Introduction.

4.

5.

¦ *) =

D(b) =

0

1

= 1 + Ux

if 6 = a,

if 6 = a + 1,

if b>a + 2,

6.

D{0,n - 1)_

?)@,n)~

i.e.

e° + exp(-(rB_i - Tn_2)) + expHT,,.! - rn_3)) +

+ exp(-(r,_i - To)) = D{n)e-T"-> (n = 1,2,...).

263

264 III RANDOM WALK IN RANDOM ENVIRONMENT

7.

(n = 1,2,...). Caution: D{n) = D{0,n); however, D(-n) ? D(-n,O).

8. I(t) is the inverse function of D(n), i.e.

I[t) = k if D{k) <t<D{k + 1),

I{-t) =ki( D{-k) < t < D(-k - 1) (t > 1; k = 1,2,...).

9. Rq, Ru ... is a random walk in random environment (RWIRE); see Intro-

Introduction.

10. p(a, 6, c); see Lemma 27.1.

Chapter 25

Introduction

The sequence {Sn} of Part I was considered as a mathematical model of the

linear Brownian motion. In fact it is a model of the linear Brownian motion in

a homogeneous (non-random) environment.

We meet new difficulties when the environment is non-homogeneous. It is the

case, for example, when the motion of a particle in a magnetic field is investigated.In this case we consider a random environment instead of a deterministic one.

This situation can be described by different mathematical models.

At first we formulate only a special case of our model. It is given in the

following two steps:

Step 1. (The Lord creates the universe). The Lord visits all integers of the real

line and tosses a coin when visiting t (t = 0, ±1,±2,...). During the first six

days He creates a random sequence

? ={..., ?_2, E-i, Eo, Ei, ?2, • • •}

where ?, is Head or Tail according the result of the experiment made in t.

Step 2. (The life of the universe after the Sixth Day). Having the sequence

{..., E-2, E-i, Eo, Ei, E2,...} the Lord puts a particle in the origin and givesthe command: if you are located in t and Ei is Head then go to the left with

probability 3/4 and to the right with probability 1/4, if Ei is Tail then go to

the left with probability 1/4 and to the right with probability 3/4. Creating the

universe and giving this order to the particle "God rested from all his work which

he had done in creation" forever.

The general form of our model can be described as follows:

Step 1. (The Lord creates the universe). Having a sequence

265


266 CHAPTER 25

? = {..., E-2, E-i,E0, Ei, E2,...} of i.i.d.r.v.'s with distribution

<x} = F{x), F@) = 0,

the Lord creates a realization ? of the above sequence. (The random sequence

{..., E-2, E-\, EQ, Ei, E2, •. •} and a realization of it will be denoted by the same

letter ?.) This realization is called a random environment (RE).

Step 2. (The life of the universe after the Sixth Day). Having an RE ? the

Lord lets a particle make a random walk starting from the origin and going one

step to the right resp. to the left with probability Eq resp. \ — Eq. If the particleis located at x = i (after n steps) then the particle moves one step to the rightresp. to the left with probability Ei resp. 1 — Ei. That is, we define the random

walk Rq, Ri, ..., by iZo = 0 and

=« + l | Rn =

1 - Pf (i^n+i = i! - 1 | Rn = i,Rn-i, Rn-2, ...,Ri) = Ei. B5.1)The sequence {J?n} is called a random walk in RE (RWIRE).

A more mathematical description of this model is the following. Let {ftl5 /i,Pi}be a probability space and let

{...E-t = E.2{ui),E-i = E-i{ui),E0 = E0{ui),Ei = El{ul),E2 =

[ux e nx) be a sequence of i.i.d.r.v.'s with Pi(#i < x) = F{x){F@) = l-F(l) =

o).Further, let {fi2, 72} be the measurable space of the sequences u2 = {el5 e2,...}

where e, = 1 or e, = — l(t = 1,2,...) and J2 is the natural er-algebra. Define

the r.v.'s YX,Y2,... on fi2 by Yi{u2) = e:t(t = 1,2,...) and let Rq = 0,J?n =

Yi + Y2 + ¦- • + Yn(n = 1,2,...). Then we construct a probability measure P

on the measurable space {ft = f^ x ft2, 7 = J\ x J2} as follows: for any givenU\ G fti we define a measure PWl = P?(Wl) = Pf on J2 satisfying B5.1). (ClearlyB5.1) uniquely defines Pf on J2.) Having the measures P?((Jl)(u;1 G fti) and Pione can define the measure P on J the natural way.

Our aim is to study the properties of the sequence {Rn}- In this study we

meet two types of questions.

(i) Question of the Lord. The Lord knows ux, i.e. the sequence ?; or in other

words, He knows the measure Pf and asks about the behaviour of the

particle in the future, i.e. He asks about the properties of the sequence

{Rn} given ?.

INTRODUCTION 267

(ii) Question of the physicist. The physicist does not know wi. Perhaps he hassome information on F, i.e. he knows something on P^ He also wants to

predict the location of the particle after n steps, i.e. also wants to describethe properties of the sequence {Rn}-

A typical answer to the first type of question is a theorem of the followingtype:

THEOREM 25.1 There exist two sequences of Immeasurable functions /M =

A) < fi2) = /?(?) such that

P < m*x\Rk\ < fW a.s. (P,) B5.2)

for all but finitely many n, i.e.

p?{fi1]{?) < max \Rk\ < fi2){?)for all but finitely many n) = 1.

Since the physicist does not know the environment S he will not be satisfied with

an inequality like B5.2). However, he wants to prove an inequality like

THEOREM 25.2 There exist two deterministic sequences a^ < a^ such that

c^ </^ </<2> < a<2> a.s. (Pl) B5.3)


Having inequalities B5.2) and B5.3) the physicist gets the following answer to

his question:

THEOREM 25.3 There exist two deterministic sequences a^ < a^ such that

cH><mBx\Rk\<ag) a.s. (P) B5.4)

for all but finitely many n. Equivalently

P{aJ' < max \Rk\ < a™ for all but finitely many n}

aJ,1) < max \Rk\ < a^] for all but finitely many n) = 1} = 1.

Remark 1. The exact forms of Theorems 25.1, 25.2 and 25.3 are given in

Theorems 27.6, 27.8 and 27.9 where the exact forms of ato,fM,ag\fW are

given.

Remark 2. In the special case when

Pi(?o = 1/2) = F(l/2 + 0) - F(l/2) = 1,

the RWIRE problem reduces to the simple symmetric random walk problem.

Chapter 26

In the first six days

In this chapter we study what might have happened during the creation of the

universe, i.e. the possible properties of the sequence ? are investigated.The following conditions will be assumed:

(C.I) there exists a 0 < 0 < 1/2 such that P(/? < Eo < 1 - /?) = 1,

(C2)

EiV0 = f°° xdP^Vo < x) = I' "log-—-dF{x) = 0y-oo Jp x

where F{x) = P^Eq < x),V0 = \ogU0 and Uo = (l - Eo)/Eo,

(C.3)

0 < a2 = ExV02 = (log dF{x) < oo.

Jp \ x /

Remark 1. In case of a simple symmetric random walk (i.e. Pi(i?o = 1/2) = 1)we have Pi(C/0 = 1) = Pi(V0 = 0) = 1 and consequently (C.l) and (C.2) are

satisfied; however, (C.3) is not satisfied since E^2 = a2 = 0. We also mention

that if (C.I) and (C.2) hold and E^2 = a2 = 0 then ^(Eo = 1/2) = 1.

Remark 2. Most of the following theorems remain true replacing (C.l) by a

much weaker condition or omitting it. Here we are not interested in this type of

generalizations.

LEMMA 26.1

HmsupTn = limsupT_n = -liminf Tn = — liminf T_n ;= oo a.s. (Pi).— - n—*oo n—^00

B6.1)n—»oo

269

270 CHAPTER 26

If we assume (C.I) and (C.3) but instead of (C.2) we assume that EiV0 = m^0.Then

lim Tn = lim T_n = (sign m)oo a.5. (P^ B6.2)n—»oo n-

Tn = n + V2 + ••• + Vn,T_n = V_i + V_2 + ••• + V_n,V;- = log #,-,#,• =

A - ?,)/?, and To = 0.

Proof. B6.1) is a trivial consequence of the LIL of Hartmann and Wintner (cf.Section 4.4), B6.2) follows from the strong law of large numbers.

LEMMA 26.2

lim Din) = oo a.5. (P^ B6.3)n—»oo

' ' v '

(cf. Notation 5.).

Proof. Since

D{n) = l + Ul + UlU2 + --- + UlU2...Un-l=e° + eT>+eT* + --- + eT"->, B6.4)

B6.3) follows from Lemma 26.1.

By B6.4) we have

exp( max Tk) < D(n) < nexp( max Tk) B6.5)0<Jk<n-l

; — V ; —

^VO<Jk<n-l ; V ;

and the LIL implies

LEMMA 26.3 For any e > 0 and for any p = 1,2,... we have

max Tk < A + e:)erBnloglognI/2a.s. (Px) for all but finitely many n. B6.6)

max Tk > A - er)crBnloglognI/2 i.o. a.s. (Px), B6.7)

max Tk < n^^lognloglogn-'-logpn)� i.o. a.s. (Pj, B6.8)

max Tk > nlsJcsfl

a.s. (Pi) for all but finitely many n. B6.9)

By B6.5) we also get

D(n) < exp{(l + er)crBnloglognI/2}a.5. (P^ for all but finitely many n, B6.10)

IN THE FIRST SIX DAYS 271

D{n) >exp{(l-e)erBnloglognI/2} i.o. a.s. (Pi), B6.11)

D{n) <exp{n1/2(lognloglogn--logpn)�} i.o. a.s. (P^, B6.12)

D{n) > exp{n1/2(lognloglogn---(logpnI+e)-1}a.s. (Pi) for all but finitely many n. B6.13)

Replacing the maxi<*<n by max_n<jk<_i the inequalities B6.6) - B6.9) remain

true. Replacing D(n) by D(—n) in B6.10) - B6.13) they remain true as theyart'

-S^LpL =a a.s. (Px), B6.14)v2nloglogn

liminf maxlog D^ J\oglogn = ott/Vs a.s. (Pj, B6.15)

n—»oo 0<Jk<n y/n

D*{n)>l, B6.16)

D*{n) < n i.o. a.s. (Pi). B6.17)

Proof. Inequalities B6.6) - B6.13) are clear as they are. The following simpleanalogue of B6.5),

-^min_i(rn_i- Tk)) < D*{n) < nexp(-^min_i(rn_i

- Tk)), B6.18)

implies B6.16) and B6.17).In order to get B6.14) and B6.15) approximate the process {7^,0 < k < oo}

by a Wiener process {erW(t),0 < t < oo}. By Theorem 10.2 the process

-m\nT(W(T)-W(t))is identical in distribution to the process {|W^(OI^ ^ 0}- Hence the LIL and the

Other LIL imply B6.14) and B6.15).

LEMMA 26.4 For any e > 0 and for any p = 1,2,... we have

W < (log |?j log log |i| - - - logp_1 |i|(logp |i|I+eJa.s. (Px) if \t\ is big enough, B6.19)

I{t) > (log jt| log log |t| - - - logp |t|J t.o.a.s. (Px), B6.20)

•¦*"¦ (Pi)' B6-21)

/@>•19iglOglll a-5" (Pi) * \t\is big enough. B6.22)2G log \t\

272 CHAPTER 26

Proof. B6.19), B6.20), B6.21) resp. B6.22) follows from B6.13), B6.12),B6.11) resp. B6.10).

LEMMA 26.5

l) (*>1), B6.23)

D{-I{-t)) < t < D{-I{-t) - 1) {t > 1), B6.24)

D{n + 1) = D{n) + UXU2 ¦Un = D{n) + UnJpj~A (n = 1,2,...), B6.25)

+ ^5,) < D(W) < t, B6.26)

D{n + 1) < ^D(n) + 1, B6.27)

I{Xt + 1) > 7@ + 1 if A > i B6.28)

where 0 is the constant of (C.l).

Proof. B6.23), B6.24), B6.25) follow immediately from the definitions. B6.23)and B6.25) combined imply

This, in turn, implies B6.26). B6.27) follows from (C.l). B6.28) follows from

B6.23) and B6.27).

Chapter 27

After the sixth day

27.1 The recurrence theorem of Solomon

THEOREM 27.1 (Solomon, 1975). Assuming conditions (C.I), (C.2), (C.3)we have

P{Rn = 0 t.O.} = Pl{P?{i*n = 0 t.O.} = 1} = 1.

Assuming (C.I), (C.3) and EiV0 ^ 0 we have

Remark 1. The statement of the above Theorem can be formulated as follows:

with probability 1 (Pi) the Lord creates such an environment in which the recur-

recurrence theorem is true, i.e. the particle returns to the origin i.o. with probability1 (Pf). Before the proof of Theorem 27.1 we present an analogue of Lemma 3.1.

LEMMA 27.1 Let

p(a, b, c) = P?{min{j : j > m, R, = a} < min{j : j > m, R3 = c} \ Sm = b}

(a < b < c), i.e. p(a,b,c) = p(a,b,c,?) is the probability that a particle startingfrom b hits a before c given the environment S. Then

Especially

p(O,l,n) = 1-—-r- and p(O,n-l,n) = —

D(n)yK ' ' ;

D*(n)

273

274 CHAPTER 27

Proof. Clearly, we have

p(a,a,c)=l, p(a,c,c)=0,

p(a, 6, c) = Ebp{a, b + 1, c) + A - Eb)p{a, b-l,c).

Consequently,

p{a, b + 1, c) - p(a, 6, c) = —^"t(p(o, 6> c) ~ P(a>6 ~ 1.c))-

By iteration we get

p{a, b + 1, c) - p(a, 6, c) = UbUb-i • • • Ua+l{p(a, a + 1, c) - p(a, a, c))= UbUb.x • • • C7a+I(p(a, a + 1, c) - 1). B7.1)

Adding the above equations for 6 = a, a + 1,..., c — 1 we get

-1 = p(a, c, c) - p(a, a, c) = D(a, c)(p(a, a + 1, c) - 1),

*a>a+l>c)rl-Dh)- B7-2)

Hence B7.1) and B7.2) imply

p(a, b + 1, c) — p(a, 6, c) = --=-t rEW6_i • • • Ua+l.V{a,c)

Adding these equations we obtain

p(a, b + 1, c) - 1 = p{a, b + 1, c) - p(a, a, c)= t^-tA + Ua+l + Ua+lUa+2 + ¦¦¦ + Ua+lUa+2 ¦¦¦Ub)

_

D{a,b + 1)D{a,c)

Hence we have the Lemma.

Consequence 1.

p(ll» p@,l,*;?) =nlim A-^=1) = 1, B7.3)

P{limp(-n,-l,0;?) = 0} = 1. B7.4)


AFTER THE SIXTH DAY 275

B7.3) follows from Lemma 27.1 and B6.3). In order to see B7.4) observe

(-n,-l) 1

and apply B6.13) for D(-n).The following lemma is a trivial analogue of Lemma 3.2; the proof will be

omitted.

LEMMA 27.2 For any -oo < a < b < oo we have

P{liminf Rn = a} = PllimsupiZn = 6} = 0.n—»oo

Proof of Theorem 27.1. Assume that Rx = 1, say. Then by Lemma 27.2 the

particle returnes to 0 or it is going to +oo before returning. However, by B7.3)for any e > 0 there exists an n0 = no(e,?) such that p@,1, n) = 1 —

—^> 1 — e

if n > no. Consequently the probability that the particle returns to 0 is largerthan 1 — e for any e > 0 which proves the Theorem.

27.2 Guess how far the particle is going away

in an RE

Introduce the following notations:

M+(n) = max J?t,V ' 0<ifc<n

M~(n) = — min Rk,

M{n) = max{M+(n),M-(n)} = max \Rk\,Osifc^fl

Po = 0,

Pi = min{ifc : k > 0, Rk = 0},

= mm{k : k > ph Rk = 0},

?(n) = max?(A:,n),

u{n) = #{«: 0 < « < n - 1, RPi+i = 1}.

Observe that ?@, pn) = n.

276 CHAPTER 27

Our aim is to study the behaviour of M(n). Especially in this section a

reasonable guess will be given.Consider the simple environment when

Note that conditions (C.I), (C.2) and (C.3) are satisfied. Note also that in the

environment ? = {..., 3/4,1/4,3/4,1/4,3/4,...} the behaviour of the random

walk is the same as that of the simple symmetric random walk. For example, it

is trivial to prove that

lim sup bnM(n) = 1 a.s.

n—*oo

if ? is the given environment and bn = Bnloglogn)~1/2.One can guess that since environment {..., 3/4,1/4,3/4,1/4,...} is nearly

the typical one, M(n) will be practically n1/2 in most environments. This way

of thinking is not correct because we know that in a typical environment there

are long blocks containing mostly 3/4's and long blocks containing mostly 1/4's.Assume that in our environment

max Tk = n1/2 and - min T-k = n1/2l<Jfe<n l<*<

which is a typical situation. Then by B6.10), B6.11) and Lemma 27.1 we have

p@,1, n) = 1 - -pJ" 1 - exp(-n1/2)

and

p(-n,-1,0) =

This means that the particle will return to the origin exp(n1/2) times before

arriving n or — n. Hence to arrive n requires at least exp(n1/2) steps. Conversely,in n steps the particle cannot go farther than (lognJ.

This way of thinking is due to Sinai A982). He was the first one who realized

that having high peaks and deep valleys in the environment, for the particle it

takes a long time to go through. Clearly high peak means that T(k) is a bigpositive number for k > 0 resp. it is a big negative number for k < 0 while the

meaning of the deep valley is just the opposite.

27.3 A prediction of the Lord

LEMMA 27.3 For any environment ? we have

TE{u{n) = k} = fyEfr - E0)n-k, B7.5)

limsup,o rj^f0} u/2=l «.-• (P.) B7.6)

where un is defined in Section 27.2.

Proof is trivial.

LEMMA 27.4 For any environment ? and k = 1,2,... we have

) <k\yn} = (p@, L ( )"

W <*}= ?(

Proof is trivial.

Now we prove our

THEOREM 27.2 For any environment ? we have

<M+(pn) </(n(lognI+e) a.s. (P?), B7.9)< M-{pn) < 7(-n(lognI+e) a.s. (P?), B7.10)

max{/(n(logn)-1-e),/(-n(logn)-1-e)} < M(pn)< max{/(n(lognI+e),/(-n(lognI+e)} a.s. (P?) B7.11)


Proof. By Lemma 27.4 and B6.26) we have

/

1--

278 CHAPTER 27

\ n

< 1- 1-E0Qn

-n(logn)

E0Qn

-(logn)1+e

where

and N = N{n) = I Qn(log nI+e) .

D*{N)Let nk = 2k. Then by the Borel - Cantelli lemma we get (cf. B6.16))

a.s.

for all but finitely many k. If nk < n < nk+1 we have

M+(pn) < M+(pnk+i) <

< / fn(lognI+eJ a.s. (P^) for all but finitely many n

Hence we have the upper part of B7.9).Now we turn to the proof of the other inequality of B7.9).By Lemma 27.4 and B6.26) we have

= 1-

Hence we have B7.9) by the Borel - Cantelli lemma.

The proof of B7.10) is identical. B7.11) is a trivial consequence of B7.9) and

B7.10).In order to get some estimates of M(n) (resp. M+(n),M~(n)) the Lord is

interested to estimate pn or equivalently ?@, n). To study this problem in a more

general form we present a few results describing the behaviour of the local time

?(z,n).LEMMA 27.5 For any integer k = 1,2,... and any environment ? we have

Eo1-

D(k)

E0(l-

if 1 = 0,

Consequently

D{k)D

Eo

IK1-^)'1*'-1'2'-"

D(k)

L-\

(?=1,2,...).

,Pi) = 0} = 1 - Eo + E0p{0,l,k)

D(k)In case / = 1,2,... we get

:,Pl) = 1} = E0(l -p(O,l,k))(l - Ek)p(O,

y=o \ ^ )

E0(l-Ek) / l-^A'�Ek) ( 1-

I>(ik)I>*(ik) V !>*(*) /

A trivial calculation gives

LEMMA 27.6 (Csorgo - Horvath - Revesz, 1987). For any k = 1,2,...

B7'12)

/or any A < - log(l - ^J). Especially

^l-e*(l-2A)Observe that

and2AeA

~2 l/ 0<A<1/2-

Proof. As an example we prove B7.14). By Lemma 27.5 we have

+-;i-^*)

{

280 CHAPTER 27

Remark 1. B7.12) implies that: for any e > 0

mk > -J— exp((l - e)crBJkloglogikI/2) i.o. a.s.

and

mk< l--^-exv{-{l-e)o{2k\og\ogkI/2) i.o. a.s.

Compare these inequalities and (9.6).

LEMMA 27.7 For any A; = 1,2,... and any environment ? we have

nmU(k,Pn)-nmk

<1

-°° \ y/nak J

limsup ^P^^ = i a.5. (p?). B7.18)n-*oo ak y/2n log log n

Proof is trivial.

Now we give a somewhat deeper consequence of B7.14).

LEMMA 27.8 (Csorgo - Horvath - Revesz, 1987). For any

and any k = 1,2,..., we have

X2E? exp(A(?(A:, Pl) - mk)) = 1 + —a\ + X3ek

where5

>k\ ^ and A is a positive constant.

Proof. By Taylor expansion we get

eA D*{k) 1

D(k)h(X) + (D(k)h(X)J + n(D(k)h(X)K)

with |0| < 1, | r) |< 1, where

Consequently

f_ x(f_if A is big enough. Hence by B7.14) we have

E?expA?(*,pi) - (l + Amt + A2

A2(Z?(A:)J - i <A\\\*(D(k)L

(D(k) \l-E

Multiplying the above inequality by

A2 A3exp(-Amfc) = 1 - Xmk + —m\+r)—rn\-

one gets the Lemma.

LEMMA 27.9 Let

0<x<ak fmin|v^,-v^logfl--^Then for any k = 1,2,... and n = 1,2,... we have

,P») " nmk\ > xy/n} < 2exp

Proof. Apply Lemma 27.8 with

A = xn�/2^2 and A

Then we get

,pn) - nmk))2)

<2

and we have the Lemma.

This last inequality gives a very sharp result for ?(fc, pn) when k is not too

big. In cases where k can be very big it is worthwhile to give another consequence

of Lemma 27.6. In fact we prove

CHAPTER 27

LEMMA 27.10 For any K> 0 there exists aC = C{K) > 0 such that

C\ognD*{k)}<n-K B7.19)

Proof. Let A = Xk = ±$j. Then by B7.15) and B7.16) we get

+ CDt{k)\ogn},pn) > expBAnmfc + XCD*{k) logn)}, Pl))n exp(-2Anmfc - \CD* [k) log n)exp

1 exp (n("

which proves B7.19).A very similar result is the following:

LEMMA 27.11 For any CY > J (cf. (C.l)) we have

< exp (-{k = 1,2,. ..;n = 1,2,...).

Proof. Let A = Afc = ^*y. Then by B7.15) and B7.16) we get

> exp -J J< exp (-

exp f-

\E?(exp(\Z(k,pn)))

E?(expA?(A:,pi))|

2AeA

B7.20)

< exp ( -

Hence we have B7.20).An analogue result describes the behaviour of ?(A:,pi) when A: is a big positive

number.

LEMMA 27.12 There exist positive constants C and C\ such that

??{Z{k,Pi) > CxD*{k)\ogk | ?(*,Pi) > 0} < Cxk-2 B7.21)

and

i) > 0} < CAT2. B7.22)

Proof. Let fj.k be the number of negative excursions away from k between 0 and

P\. Clearly, we have

Pi) > 0} = p@,k- l,*)(l - p@,* - I,*)I�

Consequently

P?{nk > L | e(*,Pi) > 0} = A - (D'ik))-1I-. B7.23)

Hence

0} < P,{m* < k^D^k) \ Z(k,Pl) > 0}= 1 - A - [Dt{k))-1)k"D'^ < Ck~2

and we have B7.22). In order to prove B7.21) observe that for any 0 < 6 < e

there exists a C2 = C2{6) > 0 such that

where ^fc = 1 - Ek - 6. Hence by B7.23) we have

log A: | ?(*,pi) > 0}l?*(*) log*,m* > ^*e(*,Pi) I ?(*,Pi) > 0}!l?*(*) log^M* < Ekt(k,p!) | e(*,Pi) > 0}

log A: | ?(*,pi) > 0}

f) P?{»k < Ekl | ?(*,pi) = i}Pf(e(*,pi) > 0}<=C!D#(Jk) log Jk

which proves B7.21).In the following lemma we investigate the probability of the event that ?(k, pn)

is very small.

284 CHAPTER 27

LEMMA 27.13

- Ek p@, k - 1, k)

with some constant C > 0.

Proof. Let

l if,

max RPi+j>k,0 otherwise,

S = Srt = fo + ?i + --- + ?n-i.

Then

pas = i} = ?o(i-p(o,i

and by the Bernstein - inequality (Theorem 2.3)

Let 1 < ti < i2 < ... < is < n be the sequence of those t's for which

max Rp+i > k

and let

i/,- = e(fc,p4i+i) - ?(*,*,.) (j = 1,2,...,5),

i.e. Vj is the number of excursions away from k between pii and p,-y+i. Further,let i/;~ resp. i/t be the number of the corresponding negative resp. positiveexcursions. Then

!/,- = !/+ +1/7,

T>e{vJ =m} = (l-q)m-lq, q = (l -

and using again the Bernstein - inequality we obtain

Hence

~

< -nmk J• • • + v~ < -nmk, S < -npj• • • + v~ < -nmk, S > -u

< Cexp (-^j + Pf |i/f + i/f + • • • + I/." < -nmk, S

Since Vj > vj we have the Lemma.

Now we give an upper bound for pn.

THEOREM 27.3 For any e > 0 and for all but finitely many n we have

Clogn j^ ?>*(fc) a.5. (Pf) B7.24)Jk=-/(-n(logn)»+«) Jk=-/(n(logn)»+«)

where C is a big enough positive constant.

Proof. By Theorem 27.2 we have

/(n(tofn)»+«)

Jk=-/(-n(logn)»+«)

Lemma 27.10 and the Borel - Cantelli lemma imply

I(n(logn)l+*)

Clogn ^ D*{k).k=0 Jk=O Jk=O

Analogous inequality can be obtained for negative fc's. Hence we have B7.24).A somewhat weaker but simpler upper bound of pn is given in the following:

THEOREM 27.4 For any e > 0 and for all but finitely many n we have

/(n(lofn)»+')

Pn < n{\ognJ+e ? mk a.s. (Pf).Jk=-/(-n(logn)»+«)

286 CHAPTER 27

Proof. By B7.12), B6.23) and (C.l) of Chapter 26 for any 0 < k < /(n(lognI+e),we have

Hence)

Clogn J2 D*{k)<n{\ognJ+2< ? mk.

k=0 Jk=O

Since analogous inequality can be obtained for negative fc's we have the Theorem.

A lower bound for pn is the following:

THEOREM 27.5

pn>-maxmk a.s. (P?) B7.25)4 cA

where A = An = {k : 0 < D(k) < k^},k < C/12 and f3 is defined in (C.l) ofChapter 26.

Proof. Since

pn > max

B7.25) follows by Lemma 27.13.

Remark 2. Remark 1 easily implies that

lim maxmjk = oo a.s. (Pi).

Hence B7.25) is much stronger than the trivial inequality pn > In.

Clearly having the upper bound B7.11) of M(pn) and the lower bound B7.25)of pn we can obtain an upper bound of M(n). Similarly having the lower bound

B7.11) of M(pn) and the upper bound B7.24) of pn a lower bound of M(n) can

be obtained. In fact we have

THEOREM 27.6 Let

ft{n) = max{/(n(lognI+<),/(-n(lognI+«)},f;{n) = max{/(n(logn)-1-e),/(-n(logn)-1-e)},

/(n(logn)'+<)

g(n) = n(lognJ+< ?

hln) = —

v '4

Jk=-/(-n(lognI+«)

Then for all but finitely many n

f:(g-lH)<M(n)<f?(h-l(n)) a.s. (P,) B7.26)

where g~l{-) resp. h~l(-) are the inverse functions of g() resp. h(-).

Proof. Theorems 27.2 and 27.4 combined imply

/; < M(pn) < M(</(n)),

which, in turn, implies the lower inequality of B7.26). Similarly by Theorems

27.2 and 27.5 we get

f?(n)>M(pn)>M(h(n))and we have the upper inequality of B7.26).

Remark 3. Note that knowing the environment ? the lower and upper bounds

of B7.26) can be evaluated.

27.4 A prediction of the physicist

Having Theorem 27.2 and Lemma 26.4 the physicist can say

7(n(lognI+e) < (lognloglogn-.-logp^nOogpnI^J B7.27)and the analogue inequalities are true for M~(pn) and M(pn). Theorem 27.2

and Lemma 26.4 also suggest that B7.27) and the corresponding inequalities for

M~(pn) and M(pn) are the best possible ones. It is really so. In fact we have

THEOREM 27.7 (Deheuvels - Revesz, 1986). For any e > 0 and p = 1,2,...we have

M+{pn) < (lognloglogn-.-logp^nOogpnI^J a.s. (P)if n is big enough, B7.28)

M > (lognloglogn--logp_1nlogpnJ i.o. a.s. (P), B7.29)

~? °S n

as (P) ^ n n big enough. B7.31)log3 n

The same inequalities hold for M~(pn) and M(pn).

288 CHAPTER 27

Proof. B7.27) gives the proofs of B7.28) and B7.31). Since by Theorem 27.2

for all but finitely many n

M+{pn)>I{n{\ogn)-1") a.s. (P?)

and by B6.20)

/(n(logn)-1-') > ((logn - (l + 2er) log2 n) log2 n- • • logp+1 nJ> (lognlogjfi- • -logpTiJ i.o. a.s. (Pi)

we have B7.29). Similarly, by Theorem 27.2

M+(pn)<I(n(\ognI+g) a.s. (P?),

and by B6.21)

hence we get B7.30). Clearly, the physicist is more interested in the behaviour

of M+(n),M~(n),M{n) than those of M+(/>„), M" (/>„), M(/>n). Since pn > In

by B7.28) and B7.30) we have

THEOREM 27.8 (Deheuvels - Revesz, 1986). For any e > 0 and p = 1,2,...we have

M+(n) <(lognloglogn---logp_1n(logpnI+'J a.s. (P)if n is big enough, B7.32)

and

The same inequalities hold for M~(n) and M(n).

To get a lower bound for M+(n),M~(n) and M(n) is not so easy. However, as

a consequence of Theorem 27.6 we prove

THEOREM 27.9 (Deheuvels - Revesz, 1986). For any e > 0 we have

for any e > 0 and for all but finitely many n. The same inequality holds forAf-(n) andM{n).

Proof. Let+

0+(n)=n(logn)a+* ? mk.

Jk=O

Then by Condition (C.I), B7.12), B6.19) and B6.6)

9+(n) < ^n(lognJ+< ? e

Jk=O

< -—^n(lognJ+€/(n(lognI+e) max

x exp((l + 2e)crB(lognJ(log2 n)

<exp(logn(loglognI+2e). B7.35)

It can be shown similarly that for any e > 0

g{n) <exp(logn(loglognI+e) a.s. (Px) B7.36)

for all but finitely many n. Consequently

^) a" (Pl) B7'37)

if n is big enough. Hence by B6.22)

B7.26) and B7.38) combined imply the Theorem.

Chapter 28

What can a physicist say about the

local time ?@,n)?

28.1 Two further lemmas on the environment

In this section we study a few further properties of the environment ?. These

results are simple consequences of the corresponding results of Part I.

LEMMA 28.1 For any 0 < e < 1 and 0 < 6 < e/2 there exists a random

sequence of integers 0 < nx = n1(u;1; e, 6) < n2 = n2(oJi; ?,6) < ... such that

Tn. <-A - e)ob~l and max Tj < n\/2{log nk)~s ¦ B8.1)

where bn = b(n) = Bnloglogn)�/2. Consequently by B6.5) and B6.18)

^2~5) and

B8.2)

Proof. B8.1) follows from E.11) and Invariance Principle 2 of Section 6.3.

LEMMA 28.2 There exist a random sequence 0 < nx = n1(u;1) < n2 =

n2(oJi) < ... and two constants Cx > 0, C2 > 0 such that

Tnk > C2b~l and

( \1/2max {Ti - Tj) < Cl —-*

. B8.3)<»<><nit

" -

Vloglogn/v '

291

292 CHAPTER 28

Consequently by B6.5) and B6.18)

D{nk) > exp (C7b-l) and

max D*{j) < exp \CX \—~^ ) I . B8.4)o<.<y<nt

wy *

\ \ loglognjk / / '

Proof. B8.3) is a simple consequence of Theorem 10.5 and Invariance Principle2.

28.2 On the local time f@, n)Since ?{Q,pn) = n Theorem 27.4 and B7.36) imply


>n) < ?@,exp(logn(loglognI+e))

i.e.

('°^.) a,. (P) B80!)

for all but finitely many N.

Now we prove that B8.5) is nearly the best possible result. In fact we have


'°^,) ,-....... (P). B8.6)

Proof. Define the random sequence {Nk} as follows: let Nk be the largestinteger for which

where nk is the random sequence of Lemma 28.1. Then by B7.9)

M+(pNk) > I(Nk(\ogNk)-ll+<M) > /(iV^logiV,)-1-') + 2 > nk a.s. (P)

for all but finitely many k, i.e. ?{nk,PNk) > 0. That is to say, there exists a

0 <j =j{k) < Nk such that ?{nk, (Pj,pj+l)) = Unk,Pj+i)-^{nk,Pj) > 0. Hence

by B7.22)P?{Z{nk,{Pj,pj+l)) < n-k2D*{nk)} < Cn~k\

WHAT CAN A PHYSICIST SAY ABO UT f@, n) ? 293

and by B8.2) and the Borel - Cantelli lemma

?(»*,(P/,Pi+i)) > nl2D*{nk) > exp((l - e)ab~lk) a.s. (P)

for all but finitely many k (where j = j{k)). Consequently

pNk > P,-+i-

Pj > ?K, (p,-,P,-+i)) > exp((l - e)ab-lk) a.s. (P) B8.7)

for all k big enough. By B6.23) and B8.2)

Vfc)-A+<) < D(I(Nk(\ogNky(l+e)) + 1) < ?>(nfc) < exp(ni/2

i.e.

n*>^log2iVfc(loglogiVfc)W. B8.8)

B8.7) and B8.8) combined imply for any 6* < 1 and for all but finitely many k

PNk>exp(\ogNk{\og\ogNkY') a.s. (P). B8.9)

B8.9) in turn implies Theorem 28.2.

Theorems 28.1 and 28.2 have shown how small ?@, N) can be. Essentiallywe found that ?@,iV) can be as small as N1/Xo*XogN. In the next two theorems

we investigate the question of how big ?@, N) can be. In fact we prove

THEOREM 28.3 There exists aC = C(f3) > 0 such that

^@, N) > exp ( (l - ^-j^J log Nj i.o. a.s. (P)

where C is defined in condition (C.I).

Proof. By B7.12) we have

Eo D(j) D(j)B8-10)

Hence by Lemma 27.10 for any K > 0 there exists a C = C(K) > 0 such that

n) > CnD'U)} <n~K {j = 1,2... ,n = 1,2,...). B8.11)

Define the random sequence {Nk} as follows: let Nk be the smallest positiveinteger for which

>nfc B8.12)

294 CHAPTER 28

where {nk} is the random sequence of Lemma 28.2. Observe that by B8.4) and

B6.23) for all but finitely many k

exp (C2b-nl) < D(nk) < D(I(Nk(\ogNk)l+<)) < Nk(\ogNk)l+e a.s. (Px).

Hence

L|^^ a.s. (PO, B8.13)3 Nk

and by B7.9) for all but finitely many k

M+(pNk)<I(Nk(\ogNkI+''2)<nk a.s. (P).

ConsequentlytU,Psk) = 0 if j>nk. B8.14)

By B8.10), Lemma 27.10 and B8.13) for any K > 0 there exists a C = C(K) > 0

such that

)*(j) J < nkNkK < ±^-^Nk-«. B8.15)

Hence by the Borel - Cantelli lemma, B8.13), B8.14), B8.15) and B8.4) we get

< CNknkexV L fp-^-H <7*^

x

VloglognJ J~

C\7*^J C\ \og3Nk

C%r log2 Nk (d \ogNk\Nexv{)

for all but finitely many k. Since similar inequality can be obtained for the sum

Ef=i Z{-J,PNk) and for pNk = E^_oo ZU>PNk) we have

if k is big enough, which implies Theorem 28.3.

Looking through the above proofs of Theorems 28.2 and 28.3 one can realize

that somewhat stronger results were proved than stated. In fact we have proved

WHAT CAN A PHYSICIST SAY ABOUT ?@, n) ? 295

THEOREM 28.4 For almost all environment ? and for all e > 0 and C bigenough there exist two random sequences of positive integers

nx = n\{?,e) < n2 = n2(?,e)... and

mx = mx(?,C) < m2 = m2(?,C) < ...

such that

?@,nfc) <exp

and

«„,«,)> op ((l-j^Jlogm,).Remark 1. Theorems 28.1 - 28.3 are, as we call them, theorems of the physicist.However, Theorem 28.4 can be considered as a theorem of the Lord. Knowingthe environment ? the Lord can find the time-points where ?@, •) will be very

big or very small while the physicist can only say that there are infinitely many

points where ?@, •) takes very big resp. very small values but he does not know

the location of these points.In the last theorem of this chapter we prove that ?@, n) cannot be very close

to n, i.e. Theorem 28.4 is not far from the best possible one.

THEOREM 28.5 For any C > 0 we have

?@, n) < exp((l - 6n) log n) a.s. (P)


Proof. Introduce the following notations:

M+{pi,Pj+i) = max Rk G = 1,2,...),

tfj*(N) = max{n : 0 < n < N,T{n) < -cri;1},n),Af+(p,-_1,p,-) > x}.

Note that by Theorem 5.8 (especially Example 3) and by Strong Invariance

Principle 2 we obtain

max Tk < e(b(*l;*(N)))-1 a.s. (P) B8.16)

296 CHAPTER 28

for any e > 0 and for all but finitely many n. Hence by Lemma 27.1, B6.5) and

B8.16)

> nexp(

Consequently if C is big enough then for any n = 1,2,... we have

and by the Borel - Cantelli lemma

J",n) a.a. (P) B8.17)

for all but finitely many n. Applying Lemma 27.1 and the definition of ij)*(N)we obtain

-O < exp (-|Hence by Lemma 27.1, B8.17) and B8.18) we get

n

^j

(JV) - l)))�) . B8.19)

Applying Theorem 5.3 we obtain

for all but finitely many N. B8.19) and B8.20) imply that with some C > 0

n > ffi^le(o,n)exp(^F@*(iV)))-1) > e@,n)exp^j a.s. (P)


Hence we have Theorem 28.5.

Remark 2. Since

there is an essential gap between the statements of Theorems 28.3 and 28.5.

Chapter 29

On the favourite value of the RWIRE

In this chapter we investigate the properties of the sequence ?(n) = max* ?(k, n).A trivial result can be obtained as a

Consequence of B7.32). For any e > 0 we have

a.s. (P). B9.1)

We also get

Consequence of B7.33).

B9.2)

It looks obvious that much stronger results than those of B9.1) and B9.2) should

exist. In fact we prove (Theorem 29.1) that (under some extra condition on ?)

^0 a.s. (P).pn—*oo Tl

THEOREM 29.1 (Revesz, 1988). Assume that

Pl(Ei = p) = ^{Ei = 1 - p) = \ @ 0 such that

limsupn-1^(n) > g(p) a.s. (P).

297

298 CHAPTER 29

Remark 1. Very likely Theorem 29.1 remains true replacing condition B9.3)by the usual conditions (C.I), (C.2), (C.3). Note that B9.3) implies (C.I), (C.2)and (C.3).

At first we introduce a few notations. Let N be a positive integer,

and define the random variables on

p+{N) = min{k :k>0,Tk = AiV},

p-(N) = min{k :k>0,T.k = -AiV},

/4A = -minlr* : 0 < k < pt{N)},= max{r_fc : 0 < k /i^. Continue the

definitions as follows:

aN = max{Jfc : 0 < k < pf{N),Tk = -/iNA},_

_

f maxjifc : 0 < k < aN, Tk + fxNA = AiV} if such a k exists,

max{k : k < 0, -Tk + fxN A = AiV} otherwise,

fc :aN <k<pt{N),Tk + nNA = AiV},

LN(j) = L(-A(fMN-j),(T^,r^)) =#{k:T^<k< r+,Tk = -/inA +JA},= max{ry - T{ : r^ 0, Rk = aN},GN = min{k : k > 0, Rk =

Hff = rain{k : k > F^, Rk = ^ or riJ} ~ Fn-

For the sake of simplicity from now on we assume that t# < 0.

The above notations can be seen in Fig. 1, where instead of the process Tkthe process

Tk =

is shown.

_

f Tk if k > 0,~

\ -Tk if k < 0

ON THE FAVOURITE VALUE OF THE RWIRE 299

Figure 1

Now we present a few simple lemmas.

LEMMA 29.1 There exists an absolute constant 6 @ < 0 < 1) such that

where

Proof. Consequence 1 of Section 13.3 easily implies that

Ti{LN(j) < 6/2 + Aj + 2,y = 0,1,..., N - 1} B9-4)is larger than an absolute positive constant independent from N. It is easy to

see that

?-,/-(*)<-} B9.5)is larger than an absolute positive constant independent from N (cf. Consequence4 of Section 10.2) and the events involved in B9.4) and B9.5) are asymptoticallyindependent as N —> oo. Hence we have Lemma 29.1.

Let N = Nk(?) be a sequence of positive integers for which

LN{J) < 6/2 + Aj + 2 (j =0,l,...,iV-l), and A holds

(by Lemma 29.1 for almost all ? there exists such an infinite sequence).

300 CHAPTER 29

LEMMA 29.2 For almost all ? and for any e > 0 we have

?@, FN) < exp ({1+

Proof. By Lemma 27.1, B6.5) and the definition of N = Nk we have

M >aN} = E0{l- p@,1, aN)) =°

JU{aN)

> —exp(— max Tk) > —exp(-

and

, HN + Fs) > exp((l -e)N),FN< exp (^ + *)N) , B9.7)

B9.8)

(N = Nk) a.s. (Pf) for all but finitely many k and

= o{GN) a.s. (Pf). B9.9)

< A - Eo) exp(- max Tk) = A -

(cf. the notations in Section 27.2). Hence by the Borel - Cantelli lemma we

easily obtain B9.6). The above two inequalities also imply that more excursions

are required to arrive at p~[(N) than at a#. Hence we get B9.9). The first

inequality of B9.7) and B9.8) can be obtained similarly. In order to prove the

second inequality of B9.7) observe that by B9.8) and B9.9)

^=E^W= E aJ,FN)<(aN-l-pj(N))exp((l + e)-)i=-oo i=r7(N)

a.s. (Pf) for any e > 0 and for all but finitely many k. Hence the lemma is

proved.

Introduce the following further notations:

Pi = Pi{<*n) = min{n : n > 0,RpN+n = aN},

h = h{aN) = min{n : n > pi,RFf/+n = as},---

U = ?{j,pn) = ?{j,FN+pn) -

aN-2

= 1+ E *xp{-{Tail-i-Tk))

and

D*{j,N) = {1-pUJ+U*n))-1 = D{j,aN).Observe that

p{j,aN-l,aN)= Dh, as) — Dlj, a^

— 1)= rr:

1~P(J,J+ !

= Uj+lUj+2 ¦ ¦ ¦ ?/„„_! = exp(Taw_i - Tj).

Clearly Lemmas 27.11 and 27.10 can be reformulated as follows:

LEMMA 29.3 For any j < an we have

P. U(j,Pn) > Of^j < exp (--^—) B9.10)

where Cx > 2p~l. Further, for any K > 0 there exists aC = C(K) > 0 such that

Pf \iU,Pn) > 2n1~/°"^(-7^) + CD*(j,N) lognl < n~K. B9.11){ tj D{j,N) J

Proof of Theorem 29.1. In order to simplify the notations from now on we

assume that r^ > 0. The case t^ < 0 can be treated similarly.Let 1/2 + e < ^i A < V>2 < 1 — ? with some e > 0 and introduce the following

notations:

n = [exp(V>2iV)] where N = Nk = Nk{?)and

X{j) = min{fc : r^ < k < aN,Tk = -/

Consider any integer / E (x{4>iN),aN). Then

( <aw(, Ta^i)) Np(ViAiV)

302 CHAPTER 29

a.s. (Pf) for all but finitely many k and by B9.10)

<expl-expl n

Consequently by B9.9) and Lemma 29.1

aN-l f)*E «y.A.)<c,»

) i

OtN-1

oo

2+ 2J exp(-/A) = /(A)n a.s. (P) B9.12)

j=o


Let / e {tn,x{1>iN)). Then by B9.11)

g + CD*(l,N) logn) < n^.

Consequently

() () ,

E e(/,Pn)<2(l-f;aN)n j: fJ^r + ClognED{lN) l=T-

2 ^naNexp(-V>iAiV) + C(logn)aNexp ( —) = o(n). B9.13)p \ z /^(VAiV) C(l) (

p

B9.12) and B9.13) combined imply

aN-\

?(J,P»)<2/(A)n a.s. (P) B9.14)

for all but finitely many k. Similarly one can see that

Hence by B9.7)

+00

J=-0O T^

Let m = 4/(A)n. Then applying again B9.7) we get

h e)m) > ?(Fn + m) > ?(Fs + pn) > ?(<*#»-FV + pn) = n =

4/(A)

which proves the Theorem.

Note that we have proved a stronger result than Theorem 29.1. In fact we

have

THEOREM 29.2 For almost all environment there exists a sequence of positiveintegers nx = nx{?) < n2 = n2(?) < ... such that

provided that the condition of Theorem 29.1 is fulfilled.

Remark 2. On the connection of Theorems 29.1 and 29.2 the message of Remark

1 of Section 28.2 can be repeated here as well.

Another simple consequence of the proof of Theorem 29.1 is

THEOREM 29.3 Assume that the condition of Theorem 29.1 is fulfilled. Then

there exists an e = e(p) > 0 such that

n—*oo \ n f

On the liminf behaviour of ?(n) we present only a

Conjecture.

lim inf -^—- log log n = 0 a.s. Pn-»oo n

and

liminf—^—(loglognK = oo a.s. P.n—oo n

Chapter 30

A few further problems

30.1 Two theorems of Golosov

Theorems 27.8 and 27.9 claim that M{n) ~ (lognJ. As we have already men-

mentioned, this fact was observed first by Sinai A982). The result of Sinai suggestedto Golosov to investigate the limit distributions of the sequences

K=cr2(lognJ

and

*""

cr2(lognJ

(for the definition of a, cf. (C.3) of Chapter 26). In order to study the limit

distributions of R^ and M^ he modified a bit the original model. In fact he

assumed that Eo = 1, i.e. the random walk is concentrated on the positive half-

line. Having this modified model he proved that the limit distributions of the

sequences {i2^} and {M^} existed and he evaluated those. In fact we have

THEOREM 30.1 (Golosov, 1983). For any u > 0

where

lim T{R*n < u} = - /U h2(v)dv C0.1)n—>oo 2./0

l *\ j Bn--zlvj and zn = ±

n=0

and

lim P{M^ <u}= fU hx{v)dv C0.2)

305

306 CHAPTER 30

where

n=0

Considering the original model Kesten A983) proved

THEOREM 30.2

2 fulim P{-Rn < u} = - h3(v)dv

n—»oo 7T J—oo

where

Sinai A982) also proved that there exists a sequence of random variables ax, a2,...

defined on Qt such that Rn —

an = o((lognJ) in probability (P). This means that

knowing the environment ? we can evaluate the sequence {an} and having the

sequence {an} we can get a much better estimate of the location of the particleRn than that of Theorem 27.6. Golosov proved a much stronger theorem. His

result claims that Rn —

an has a limit distribution (without any normalising fac-

factor), which means that knowing an the location of the particle can be predictedwith a finite error term with a big probability. The model used by Golosov is

a little bit different from the one discussed up to now. He considered a random

walk on the right half-line only and he assumed that the particle can stay where

it is located. His model can be formulated as follows.

Let ? — (p-i(n),po(n),p1(n),} (n = 0,1,2,...) be a sequence of indepen-independent random three-dimensional vectors whose components are non-negative and

p_i@) = 0, p-i(n) + Po(rc) + Pi(rc) = 1 (n = 0,1,2...). Assume further that

(i) (p-i(n),pj(n)) (n = 1,2...) are identically distributed,

(ii) po(n) (n = 0,1,2,...) are identically distributed,

(iii) the sequences {po{n), n = 0,1,2,...} and {p-i(n)/pj(n), n = 1,2,...} are

independent,

(iv) Elog(p_1(n)/p1(n)) = 0 and 0 < E(log(p.1(n)/p1(n))J = a2 < oo,

(v) E(l - po(n))� < oo and P(po(n) > 0) > 0.

Having the environment ? we define the random walk {-Rn} by Rq =0 and

(p-^i) if 9 =-I,

Y?{Rn+i = i + 0\Rn = i,Rn-U---,Ro} = { Po@ if 0 = 0,

(pi(i) if 0=1-

Then we have

THEOREM 30.3 (Golosov, 1984). There exists a random sequence {an} de-

defined on Qi such that for any— oo < y < oo

lim T{Rn -an<y} = F{y)n—»oo

where the exact form of the distribution function F(y) is unknown.

Remark 1. Clearly Theorems 30.1 and 30.2 can be considered as theorems

of the physicist. However, Theorem 30.3 is a theorem of a mixed type. The

physicist knows about the existence of an but he cannot evaluate it. The Lord

can evaluate an but He cannot use His further information on ?. In fact, He

would like to evaluate the distribution Pf {-Rn —

<*n < */}¦ I* is not clear at all

whether the \\mn-,0O'P?{Rn —

an < y} exists for any given ?.

Remark 2. Theorems 29.1 and 29.2 also suggest that Rn should be close to an.

30.2 Non-nearest-neighbour random walk

The model studied in the first five chapters of this Part is a nearest-neighbourmodel, i.e. the particle moves in one step to one of its neighbours. In the last

model of Golosov the particle keeps its place or moves to one of its neighbours.In a non-nearest-neighbour model the particle can move farther. Such a model

can be formulated as follows.

Let ? = (P-i(rc),Pi(n),P2(rc)} {n — 0,±1,±2,...) be a sequence of inde-

independent, identically distributed three-dimensional random vectors whose com-

components are non-negative and p_i(n) + Pi(n) + p2(n) = 1 (n = 0,±l,±2,...).Then we define a random walk {Rn} by Rq = 0 and

pi[i) if 0 = 1,

P2{i) if 9 = 2.

Studying the properties of {Rn} is much, much harder than in the nearest-

neighbour case. Even the question of the recurrence is very hard. In fact, the

question is to find the necessary and sufficient condition for the distribution

function

which guarantees that

T{Rn = 0 i.o. } = 1. C0.3)

308 CHAPTER 30

This question was studied in a more general form by Key A984), who in the

above formulated case obtained the required condition.

THEOREM 30.4 (Key, 1984). Let

<* = Pi@) + P2@) + ((Pl@) + P2@)J + 4p_1@)p2@))J/2Bp_1@))-1

and

(y)Then

P{Rn = 0 i.o.} = 1 if m = 0,

P{ lim Rn = oo} = 1 if m > 0,

P{Jiin Rn = -oo} = 1 if m < 0.vn—»oo

Remark 1. Clearly this Theorem gives the necessary and sufficient condition

of C0.3) if the expectation m exists. If m does not exist then the necessary and

sufficient condition is unknown, just as in the nearest-neighbour case.

Remark 2. The general non-nearest-neighbour case (i.e. when the environment

? is defined by an i.i.d. sequence

where L and R are positive integers) was also investigated by Key. However, he

cannot obtain an explicit condition for C0.1), but he proves a general zero-one

law which implies that

P{Rn = 0 i.o. } = 0 or 1.

His zero-one law was generalized by Andjel A988).

30.3 RWIRE in Zd

The model of the RWIRE can be trivially extended to the multivariate case.

For the sake of simplicity here, we formulate the model in the case d = 2. Let

Un = {UJ}\ulp,U§\ufp) {i,j = 0,±l,±2,...) be an array of i.i.d.r.v.'s with

\? > 0, ?#> + UW + UV + UW = 1. The array ? = {UiJt ij = 0, ±1, ±2,...}

is called a two-dimensional random environment. Having an environment ? a

random walk {Rn, n = 0,1,2,...} can be defined by Rq = 0 and

P{i2n+1 = (», j + l)\Rn = {iJ),Rn-u- --,

P{Rn+1 = (i-hj) I Rn = (i,j),Rn-x,.-.,Ro} = U$\1

= [ij ~l)\Rn = {i,j),Rn-U ¦ ¦ , ^No non-trivial, sufficient condition is known for the recurrence

P{i2n=0 i.O.}=l-

in the case d > 2. Kalikow A981) gave necessary conditions. In fact, he gave a

class of environments where P{-Rn = 0 i.o.} = 0. As a consequence of his result

he proves

THEOREM 30.5 Define the environment ? by

and

Pij^.^.^.^) = {<>2,l>2,c2,d2)} = l-p.

Assume that

p[a\ — C\) I a\ b\ C\ d\\rK ^—r- > max —.-r-.—>T" •

\\ — p)[c2 — a2) \a2 b2 c2 d2)Then

Y{Rn = 0 i.o. } = 0;

moreover

lim RW = oo a.s. (P)

where R^ is the first coordinate of Rn.

Kalikow also proves a zero-one law, i.e. he can prove under some regularityconditions that P{-Rn = 0 i.o.} = 0 or 1. This zero-one law was extended byAndjel A988).

Kalikow also formulated some unsolved problems. Here we quote two of them.

Problem 1. Is every 3-dimensional RWIRE transient?

310 CHAPTER 30

Problem 2. LetO<p<l/2 and define the random environment ? by

Is this RWIRE recurrent?

30.4 Non-independent environments

In Chapter 25 we mentioned the magnetic fields as possible applications of the

RWIRE. However, up to now it was assumed that the environment ? consists

of i.i.d.r.v.'s. Clearly the condition of independence does not meet with the

properties of the magnetic fields and most of the possible physical applications.In most cases it can be assumed that the environment is a stationary field. A

lot of papers are devoted to studying the properties of the RWIRE in case of a

stationary environment ?.

In the multivariate case it turns out that having some natural conditions

on the stationary environment ? (which exclude the case of independent en-

environments) one can prove the recurrence and a central limit theorem with a

normalizing factor (lognJ.

30.5 Random walk in random scenery

Let a = (a, = o(i), i = 0, ±1, ±2,...) be a sequence of i.i.d.r.v.'s with

Ea, = 0, Ea? = 1, E(expt<7.) < oo

for some \t\ < t0 (t0 > 0). a is called random scenery. Further, let {5*} (in-(independent from {ak}) be a simple symmetric random walk. Kesten and SpitzerA979) were interested in the sum

fc=0

If the particle has to pay $ a, whenever it visits i, then the amount paid by the

particle during the first n steps of the random walk is n3lAKn. Clearly

+00

fc=-oo

where ?(•, •) is the local time of {Sk}-Studying the sequence {Kn} Kesten and Spitzer are arguing heuristically as

follows: let T/Uo a* = Lh t^ien one can define independent Wiener processes Wxand W2 such that W2(n) should be near enough to Ln and simultaneously ?(k, n)should be near to the local time rji(k,n) of the Wiener process Wi(-). Hence

Kn~n-Z'A ? (W2(k + l)-W2(k))Vl(k,n)k=-oo

+oo

rn[x,n)dWt{x). C0.4)r+oo

J — OO

Since it is not very hard to prove that n 3/4 J+~ rji(x, n)dW2{x) has a limit dis-

distribution, the above heuristic approach suggests that Kn has a limit distribution.

Applying Invariance Principle 2 (Section 6.2) and Theorem 10.1 it is not

hard to get a precise form of C0.4).We note that Kesten and Spitzer investigated a much more general situation

than the above one and they initiated an extended research of random sceneries.

As an example we refer to Bolthausen A989) where a multivariate version of the

above problem is treated.

30.6 Reinforced random walk

Construct a random environment on R1 by the following procedure. Let Rq =

0, P{.Ri = 1} = Y{Ri = -1} = 1/2 and let the weight of each interval (»,» +

1) (t = 0,±1,±2,...) be initially 1 and increased by 1 each time the process

jumps across it, so that its weight at time n is one plus the number of indices

k < n such that (Rk,Rk+1) is either (»,» + 1) or (» + 1,»). Given {Ro = 0, Rx =

»i, ..., Rn = in} Rn+l is either tn + 1 or tn- 1 with probabilities proportional

to the weights at time n of (in,in + 1) and (tn — l,tn) where »i,»2, •••,»'* is a

sequence of integers with |tJ+i — ij\ = 1 {j = l,2,...,n). Hence if i2i = 1, the

weight of [0,1] at time n = 1 is 2. Consequently

= - and -P{R2 = 0 | #i = 1} = ?.3

Similarly

P{i22 = -2 | i2j = -1} = i and P{R2 = 0 | R, = -1} = |.Further, in the case Rx = 1, R2 = 2 the weights of [0,1] and [1,2] at time n = 2

are equal to 2. Hence

1T{R3 = 3\Rl = l,R2 = 2} = 1 - P{i23 l\R1 l,R2 2}

312 CHAPTER 30

In the case Ri = 1, R2 = 0 the weight of [0,1] at time n = 2 is equal to 3. Hence

P{i23 = 1 | Ri = 1,R2 = 0} = 1 - P{i23 = -l\R1 = l,R2 = 0} = -.

Similarly

P{i23 = -3\Rl = -l,R2 = -2} = 1 -P{i23 = -l\Ri = -l,R2 = -2} = \and

P{i23 = -1 | Rx = -l,R2 = 0} = 1-P{i23 = 1 | Ri = ~l,R2 = 0} = -.

This model was introduced by Coppersmith and Diaconis (cf. Davis A989))and studied by Davis A989, 1990).

Intuitively it is clear that the random walk generated by this model is "more

recurrent" than the simple symmetric random walk. However, to prove that it

is recurrent is not easy at all. This was done by Davis A989, 1990), who studied

the recurrence in more general models as well.

Note that in this model the random environment is changing in time and

depends on the random walk itself. Situations where the random environment is

changing in time look very natural in different practical models.


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A981) Generalized random walk in a random environment. The Annals of Prob-


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A988) Maximal length of common words among random letter sequences. The


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Author Index

Andjel, E. D. 308, 309

Auer, P. 217, 222, 233, 235, 259

Bartfai, P. 53, 74

Bass, R. 130, 131

Berkes, I. 32

Bernoulli, J. xiii

Bickel, P. J. 170

Billingsley, P. 16

Bingham, N. H. 38

Bolthausen, E. 88, 311

Book, S. A. 67

Borel, E. 29

Borodin, A. N. 103

Brosamler, G. A. 146, 147

Chen, R. W. 59

Chung, K. L. 108, 114, 116, 142

Coppersmith, N. 312

Csaki, E. v, 18, 41, 44, 68, 70, 77, 78,84, 89, 90, 103, 108, 109, 110,111, 114, 115, 116, 117, 118,119, 120, 122, 123, 136, 137,139, 143, 144, 145, 157, 162,163, 164, 166, 169

Csorgo, M. v, 63, 69, 115, 116, 122,133, 134, 136, 137, 147, 175,279, 280

Darling, D. A. 170

Davis, B. 312

De Acosta, A. 89

Deheuvels, P. 18, 61, 71, 72, 75, 76,287, 288

Devroye, L. 76

Diaconis, P. 312

Dobrushin, R. L. 124, 125

Donsker, M. D. 209

Dvoretzky, A. 187, 195, 197, 207, 215

Erdos, P. v, 17, 18, 35, 39, 53, 57, 59,

62, 64, 74, 75, 116, 129, 130,142, 143, 144, 145, 170, 184,187, 195, 197, 200, 202, 205,206, 207, 213, 214, 215, 217,242

Feller, W. 19, 20, 35

Fisher, A. 146

Foldes, A. v, 18, 21, 70, 77, 78, 103,

114, 115, 116, 117, 118, 119,122, 123, 136, 137, 139, 145,160, 162, 163, 164

Gnedenko, B. V. 19

Golosov, A. O. 305, 306, 307

Goncharov, V. L. 21

Goodman, V. 90

Griffin, P. 130, 131, 186

Grill, K. 18, 44, 65, 66, 68, 75, 88,166, 168, 169

Guibas, L. J. 57, 59

Hanson, D. L. 69, 171

Horvath, L. 279, 280

Hunt, G. A. 108, 114

Imhof, I. P. 165

327


328 AUTHOR INDEX

Ito, K. 147, 173

Kakutani, S. 215

Kalikow, S. A. 309

Karlin, S. 63

Kesten, H. 108, 114, 259, 260, 306,

310, 311

Key, E. S. 307, 308

Khinchine, A. 30

Knight, F. B. 107, 145, 190, 191, 242

Kolmogorov, A. N. 19, 25, 35, 38

Komlos, J. 18, 53, 54

Kuelbs, J. 90

Kusolitsch, N. 63

Lacey, M. T. 147

Lamperti, J. 256

Lawler, G. F. 214

Le Gall, J. F. 211

Levy, P. 33, 100, 107, 146, 147

Lynch, I. 76

Major, P. 53, 54, 132

McKean Jr, H. P. 147, 175

Mogul'skii, A. A. 112

Mori, T. 59, 60

Mueller, C. 91, 121

Nemetz, T. 63

Newman, D. 132

Odlyzko, A. M. 57, 59

Orey, S. 194

Ortega, I. 64, 68

Ost, F. 63

Pascal, B. xiii

Perkins, E. 147

Petrov, V. V. 63

Petrowsky, I. G. 35

Philipp, W. 147

Polya, G. 23, 183

Pruitt, W. E. 194

Puri, M. L. 160

Quails, G. 170, 173

Renyi, A. 14, 19, 20, 28, 64, 74, 97

Revesz, P. 17, 18, 39, 57, 60, 62, 63,67, 68, 69, 71, 72, 75, 84, 87,105, 108, 109, 110, 111, 115,116, 120, 122, 129, 130, 133,134, 136, 137, 143, 144, 147,162, 163, 164, 175, 217, 218,227, 233, 235, 238, 241, 250,279, 280, 287, 288, 297

Riesz, F. 81

Rosenblatt, M. 170

Russo, R. P. 69, 171

Samarova, S. S. 57

Schatte, P. 146

Shore, T. R. 67

Simons, G. 109, 111

Sinai, JA, G. 276, 305, 306

Skorohod, A. V. 52

Solomon, F. 273

Spitzer, F. 27, 191, 192, 219, 242,310, 311

Steinebach, J. 76

Strassen, V. 32, 86

Szabados, T. 173

Szekely, G. 18

Sz. - Nagy, B. 81

Taylor, S. J. 145, 184, 195, 197, 200,

202, 205, 206, 213, 214, 215,242

Toth, B. 260

Trotter, H. F. 101

Tusnady, G. 18, 53, 54

Van Zwet, W. R. v

Varadhan, S. R. 209


AUTHOR INDEX 329

Varga, T. 55, 58

Vincze, I. 109

Watanabe, H. 170, 173

Weigl, A. 147

Wichura, M. 91

Wschebor, M. 64, 68

Zimmermann, G. 133


Subject Index

Arcsine law 100

Asymptotically deterministic sequence

34

Bernstein inequality 13

Borel - Cantelli lemma 27

Brownian motion 9

Chebyshev inequality 28

Dirichlet problem 255

DLA model 258

EFKP LIL 35

Gap method 29

Invariance principle 52, 105, 189

Ito formula 173

Ito integral 173

Large deviation theorem 19

Levy classes 33

LIL of Hartman and Wintner 32

LIL of Khinchine 31

Logarithmic density 146, 205

Markov inequality 28

Method of high moments 29

Normal numbers 29

Ornstein - Uhlenbeck process 169

Percolation 259

Quasi asymptotically deterministic se-

sequence 34

Rademacher functions 9, 11

Random walk in random environment

definition 265

local time 278, 292, 297

maximum 275, 277, 287, 305

recurrence 273

Random walk in random scenery 310

Random walk in Zx

definition 9

excursion 143, 147, 152

favourite values 129

first recurrence 94

increments 17, 73

increments of the local time 118

law of the iterated logarithm 31

law of the large numbers 28

local time 95, 100, 105, 113, 122,152

location of the last zero 98, 100,

141, 165

location of the maximum 98,100,142, 161

longest run 21, 55

longest zero-free interval 141

maximum 14, 20, 31, 35, 41

maximum of the absolute value

17, 20, 31, 35, 41, 163

mesure du voisinage 148

number of crossings 96, 109

range 44

rarely visited points 132

recurrence 23

Strassen type theorems 80, 86, 88,120

Random walk in Zd

completely covered balls 241

completely covered discs 217

definition 181

331


332 SUBJECT INDEX

favourite points 232, 247 law of the iterated logarithm 193

first recurrence 197 maximum 192

law of the iterated logarithm 193 rate of escape 195

local time 197, 231 selfcrossing 213

maximum 192 Strassen type theorems 194

range 207 Wiener sausage 210

rate of escape 195 Wiener sheet 133

recurrence 183 Zero-one law 25

selfcrossing 213

. speed of escape 249

Strassen type theorems 194

Reflection principle 15

Reinforced random walk 311

Skorohod embedding scheme 52

Tanaka formula 175

Theorem of Borel 28

Theorem of Chung 39

Theorem of Donsker and Varadhan

121

Theorem of Hausdorff 30

Theorem of Hirsch 39

Wichura's theorem 90

Wiener process in Rldefinition 48

excursion 145, 147

increments 63

increments of the local time 115

local time 100, 105, 187

location of the last zero 169

location of the maximum 165

longest zero-free interval 116

maximum 53

maximum of the absolute value

53

mesure du voisinage 147

occupation time 100

Strassen type theorems 80, 86, 88,120

Wiener process in Rddefinition 189


random walk in random and non-random environments - by pal revesz

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