THEORY OF PROBABILITYVoum W AND ITS APPLICATIONS Number

1961

MORE EXACT STATEMENTS OF LIMIT THEOREMS FORHOMOGENEOUS MARKOV CHAINS

S. V. NAOAEV

(Translated by M. Petrich)

Introduction. Formulation of Results

Theorems which give more exact statements about convergence to thenormal law in the case of homogeneous Markov chains with a finite number ofstates have been first obtained by S. Kh. Sirazhdinov 2. In the present paperthese results are extended to the case of an arbitrary number of states (twotheorems of this type have been proved in _6). Moreover, we prove theoremsanalogous to theorems of Cram6r 4 and Richter (5 Theorem 3).

Thus let X be a space of points , x the a-albegra of its subsets, p (, A)the transition probability function. For a fixed , p (, A is a probability meas-ure and, for a fixed A, p (, A) is measurable with respect to x.

Let p(, A) satisfy the following condition: there exists a positive integer kosuch that

(0.1) sup I#I01 (L A)--f<eo)(W, A)[ < 1, A e x, , 7 e X,,,A

where p(ol (, A) is the transition probability/unction/or ko steps.If condition (0.1) is satisfied, then there exists a stationary distribution

pCA) such that

(0.2) sup IP (A)--p(")(, A)I --<-- [-/o] < yp,,

where , 6-, p 6/0. From now on we shall always assume that condition(0.1) is satisfied.

Consider the sequence of random variables x, x.,.-., x,,..., defined asfollows:

P (x e A) z (A),(o.)

P(x A) / p(n-1)(, A)z(de)..Ix

Let/() be a real function defined on X and measurable with respect to x.If

z(A p(A and J’x/()p(d)then there exists

(0.4) lim M1

(/(x,)-- /()p(d7) a O.

62

Limit theorems /or homogeneous Markov chains 63

Let us denote by F,(x) the distribution function of the normed sum

and by qg(x) the normal distribution function.

Theorem 1. I/there exists a ]unction g(x) > 0 o] a real variable such that

(o.)

then

lim g (x c

sup fx II(’)lg(ll(’)llP(’ dn) < co,

xl/()l(d) < m and > O,

[F,(x)--qb(x)[ < c1 aa%/ +c.a +ca + - p 1+

uni]orraly in x, where

ma sup fxc(p) [136 Ma(p)+-a-M(p)+l],

M(e) [2.3oko(l+2p)+3];

q, c., ca are constants independent o/p(, .) and z(.).We now introduce several conditions which appear in different combina-

tions in the theorems formulated below.

CONDITION (A). There exists a C x such that p(C) > 0 and 0 < m <Po(, ) < M < o /or , C, where Po(, ) is the density o/the componentp(, .) which is absolutely continuous with respect to p(.), and moreover, Po(, )is measurable with respect to x x.

CONDITION (Bk). There exists a /unction g(x) o/ a real variable such that

and

lim g(x) oo

CONDITION (C). Let X be a countable set o/states i which/orms a positiveclass,/(i) a+kih, where ki is an integer, a is any real number, h > O, and/orarbitrary i and it is possible to lind an index k such that )ik > 0 and

CONDITION (D). There exist constants A > 0 and M > 0 such that

l.fxe1(’)P(, d) < M,

uni/orraly with respect to X and Izl A.

<M

64 S. V. Nagaev

Note that if one of the conditions (A) and (C) is satisfied, then a > 0.

Theorem 2. I/conditions (A) and (Bk) are/ul/illed/or k 3, the distri-bution F(x) P( /() < x} is not latticed and

then

Fn,(x)--q)(x

uni]ormly in x. Here

’where

)()

M is computed with the hypothesis that xI has stationary distribution, M,, withthe hypothesis that the initial distribution is =(.).

Theorem 3. I/conditions (Ba) and (C) are/ul/illed, the greatest commondivisor ki equals 1, and

then

F.(x)-(x)

uni/ormly in x, where

1 e_,/3 + +o

S(x)=[x_-x+-, an

QIr(X) is given as in Theorem 2.

Theorem 4. //conditions (B) and (C) are/ul/illed, the greatest commondivisor ki equals 1, and

then

rrn+l ($)

i=l

(Zns)- 1 ()m (%/.)/-1

Limit theorems 1or homogeneous Markov chains

uni]ormly in S, where

n(S) P (l /(Xi) an+sh)under the condition that the initial distribution is 7 7(i)

a/nz a(n+l)+sh--(n+l) /(i)P(i);1

Pm.(n) are polynomials o/ degree 3m, whose coe//icients depend on the initialdistribution i, and P.,.(--q) is computed like P.(--u) by substituting/or u

1 d() (x) e-.

VdxTheorem 5.2r/conditions (A)and (B)are satis/ied,

f l/()l-=() <and

then

lira inf ffc sin2(/()--/())0p(d)p(d) > O,

uni/ormly in x. Here1---e-! Qs.(x) P.(-#),

where P.(--#(x)) is computed as in Theorem 4.

Theorem 6. I] condition (D) is /ul]illed, x o(), and x > 1, then

1--Fn’(X) e(.,iV)(.IV) I+O ()-()and

where #(t) is a power series which converges/or t o/ su//iciently small absolutevalue.

Theorem 7. I/conditions (C) and (D) are/ul]illed, the greatest commondivisor ki equals 1, and IZn[ 0(/), Izl > 1 (see Theorem 4), then

where/ (t) is the series appearing in Theorem 6, q(x) is the density o] the normaldistribution.

66 S. V. Nagaev

1. Proof of Theorems 1--5

1. Let be the Banach space of bounded complex functions g(), e X,Yt* the Banach space of complex totally finite measures #(A), A e x, withthe norm II/ll I/l(X), where I#](’) is the totalvariation of the measure #(.).Define the linear operator P(O) on in the following way:

(1.1) P(O)g fx e’nP(e’ drl)"

From now on we shall denote the operator P(0) by P. Let R(z, 0)and R(z)betheresolvent operators for P(0) and P, respectively. If

then

[IP(O)-P[I <

(1.2) R(z, O) , R(z)[(P(O)--P)R(z)].0

Let I and I be circles with centers at points 1 and 0, respectively, and radiix supllR(z)Iip: (l--p)/3 and p. +p. If [IP(O)--PI[ < liMp where Mp

in the closed region which is exterior to the discs bounded by the circles 11 andI., then 11 and I. lie in the resolvent set of the operator P(O).

By Lemma 1.1 of [6] there exists an e > 0 such that for [[P(O)--PII < e

(1.3) P’(O) I.’(O)P(O)+P’(O)P(O),

where

PI(O) R(z, O)dz, P(O) R(z, O)dz,

(P(O)Px(O)% b)(o)

is a function identically equal to 1. Here and from now on, the functional.[xg()#(d) is denoted by the symbol (g, #), g e, # e*, is chosen in such a

way that for IIP(O)--PII < e the inequality []P(O)--Px]I < 1 is fulfilled,where P1 PI(0). It is not hard to see that

Pg fx g()p(d),where p(.) is the stationary distribution. From the definition of P(O) it followsthat

,(1.4) [IP(O)-PII <- llR(z, O)--R(z)lldz.

By (1.2) for z on It

,(1.5) IIR(z, O)--R(z)ll < MIIP(O)--P[I1--MpIIP(O)--PII

On the other hand,

Limit theorems [or homogeneous Markov chains 67

P (z) P1+ X(P’-- PI)z-’-l,z--1 o

(cf. [6], page 392) and consequently

(1.7) M <

Obviously

[2.3ok0(l+2p)+3] M(p).

1 1(1.8) Mg(p---- < 2M(p)

Izl < v,

From (1.4), (1.5), and (1.8) it follows that e can be put equal to 1/2M-2(p).Since

(1.9)where

I[P(O)--PI[ <= m,lO[,

m sup fx I/(V)IP(, @),

the decomposition (1.3) is true for

(1.10) IO[ <It is easy to see that

(1.11)where

2Mg(p)ml

Me’s (Pn(O)p, z(O) ),

(o, A) fa

(1.13)

Lemma 1.1. For

Consequently for IOI < 1/2M2(p)m

(1.12) nei8- 2’(O)(P(O)p, (O))+(Pn(O)Pg,(O)p, z(O)).Without loss of generality we can assume that 2’(0)--0, i.e.

f/()p() o.

the inequality

holds, where

IOl < Tn5/3

f13 (136M3 (p) z2+--M (p) + 1 )mz

PROOF. It is not hard to see (see [6], page 394) that

(P(O)Px(O)w p) 1+ B(O),1

(PI(O)v, p) 1-4- , C(O),1

68 S. V. Nagaev

where

(1.15)

and

)Be(O) zR(z)A(z, O)dz% p

C(O) R (z)A (z, O)dzv/, p

A (z, O) (P(O)--P)R(z).Note that (6], page 395)

(1.16) BI(0 Je’mlp(d)--l, C1(0) O.

Moreover, from (1.6) it follows that

P(1.17) R(z)lz

Denote (P(O)PI(O)%p) and (Pl(O)%p) by #1(0) and #.(0), respectively.From (1.13)--(1.17) we infer #i(0) 0, i 1, 2,

(1.18)

where

Furthermore

(1.19)

41--p( 6b.--[’(o)l--< u -p M

e (p) + 1) m. fl <5 4mIM (p),

Iff;(0)l < l--P( 6M. )3 ]---- (p)+l me<fie,

--p

m. sup fx I/(V)IP(#’ dr).

’"(0) follows from (0.5). By (1.13)--(1.17)The existence of _i

(1.20)

Thus, if OI < 1/2mM(p), then

(1.22) ]#;"(0)1 (136 M3(p)+ --M2(p)+ l)ma .Analogously, /or [0[ < (1/2m)M-(p)

(1.23)

Evidently

(1.24) f13 > 136Ma(p)m3

Limit theorems [or homogeneous Markov chains 69

Furthermore

(0)(’) V; -It is not hard to see that

b.02 ,,,( 0 ) 03

V /m @7 a(C)’ o=< 101_ 101.

(1.26)1

where mathematical expectations are computed under the hypothesis thatn(.)--p(.) (see [6], page 405). On the other hand,

(1.27) IM/ (x)/(x+)

(cI. [7], page 203, Lemma 7.1). Consequently,

(1.28) a < 1+4k 1--/ m. <1--p

m..

From (1.7), (1.24), and (1.28)we infer

T a2 m2 1(eg) oV <

evemi(o)<eml/()

Consequently, for 10l =< T

(0)if g < fl and fl < fl. Analogously for [0[ < T

44(.a) <

Thus for Iol < T

(1.32)

(1.33)

where

46

(1.34)

(a)= exp In (log # (r/) --log

[ (0)n log# aV, --log /*2 2

log #1

/*2 (0)J o=a/v’o < 101 < 101.

It is not hard to show that

(1.a)

From (1.33) we conclude that

(1.36) %n

I1 < 13fla.

e-212[e6*’v’-- 1 I.

70 S. V. Nagaev

Since

(1.37)

then

le-- 1I I1 exp (I//I),

<___ e-Ol2 136 -/-- exp10]a/3a (13 [0[ aa

Finally for [0[ _<_ T

13 10[ flz 130.(1.39)6 (r3/ < 3--

The assertion of the lemma follows from (1.38) and (1.39). We now continuethe proof of Theorem 1. By (1.4), (1.5), and (1.9) for

(1.40) P a/ % z (r%/ 1 =< (2mM2 (p) +111)"Since fhz R (z)dz O,

ff l fiznConsequently, for [0] < (an/2m)M-(p)

(1.41) IOl<7. 2rt,t Mg.p

__m l p

rVt

Since (ml/C)M(p) < fla/(ra, it follows from (1.12), (1.40), (1.41), and Lemma1.1 that for [0l < Tn

(1.42)

where

It remains to apply Esseen’s theorem (see [1], page 211), setting Aand T T

2. Lemma 1.2. I[ r(.)= p(.)x/()P(d)--0 and condition (Ba) is

[ul]illed, then

i-a[ da-log 2(0)o=o

M/a(x)+ 3 Z M/2(x)l(x+)

-I-3 , M/(x)]2(xe+I)-t-6 ., M/(Xl)](xe+x)/(xe++).

PROOF. Using (1.12) it is not hard to show that

Lsmit theorems [or homogeneous Markov chains 71

i- logA(0) o--o

M/()/(X,+l)/

1lira --M(l(z)+l(x)+"" +l(z))

lim Z M/(,)t(z)/().noo i,i,/

where V(, A) is the total variation of p(, .)--p(.) on A. Applying HSlder’sinequality, we successively obtain"

IM/(x)/(x,+)/(x,++)l II()lP(d)

From (0.2), (1.44)--(1.47), taking into account that V(, .) _--< p()(, ")+p(’),we conclude that

IM/(Xl)/(xi+:)/(xi++l) < 4p(ila)+(’13)V’Ml/(x) .Furthermore

n n--i n--i--It

+ , , , MI(,)I(,+)I(,++)i=1 k=l m=l

n--1 n--1

t M/a(xl)--3 M/2(Xl)/(X+l)-3 Z M/(xi)/2(xv,+l)k=l

+6 Zn--1

--3 kEM/2(Xl)/(x+i)--M/(Xl)/(Xk+l)]

-6 , (k+i)Ml(x:)l(x+:)l(x,++).k+j

_n--1

72 S. V. Nagaev

The assertion of the lemma follows from (1.43), (1.48), and (1.49).Lemma 1.3. I/ condition (Bk) is /ul/illed /or k--1, then

l fx(1.50) i-1d

(PI(0)P, y:(0))o=o

=() f()p,(, )+ f()=(d),

whr f() =/(,)--y/()(a).For the proof it is necessary to use the decomposition (1.).Lemma 1.4. I/condition (C) is ]ul/illed and the greatest common divisor

k, equals 1, then/or any e > 0 there is ce 0 such that/or n no (no does notdepend on e)

[[P"(0)[I < e-"*",i/ o (=/)-.

Lemma 1.4 has been essentially proved in 6. Indeed, from the condition

inf min(p,, p) > 0,

which was stated in the formulation of Lemma 3.1 in 6, follows the condition’./or an arbitrary k there exist i and such that p > 0 and p 0 hold simul-taneously, which is used in the proof.

Lemma 1.5. I/ condition (A) is /ul/illed, then there exists a q such that

t/,,(o)I =< aM #(C) in--2 (/()--/(Z))p()(aZ)

Poor. Let a sequence of totally finite complex measures p")(O, , A)be defined in the following way:

(0, , A) p(0, , A) eiO()p(, d),

-(0, , A) =fa p--l(0, , A)p(O, d).

Let (#,-) be the singular component of p(, .). It is not hard to see that

Consequently, the total variation of p(O, , .) satisfies the inequality

, X)G 1--rP(d:)rPo(, )Po(, )p(d)V(O,(.) X x

fxO(, )o(, )eif()(d) }

Limit theorems tot homogeneous Markov chains 73

Furthermore

ei(’)Po(, 7)Po(, )p(d)

.Ix dx

Whence by (A) for e C, e C

Po(, )Po(, )p(d) e*"Po(, )Po07, )p(d)

LL o(1.53) 2 Po($, )Po(, )Po(, 2)Po(2, ) sin- ([(rl)--/(’))p(drl)p(dX)

-:> 2m4 sin2- (l()--l(X))P(d)p(dX).

On the other hand, for e C, C

From (1.53) and (1.54) for e C, e C it follows that

(1.55) x#O(,)#o(, )p(d)-- fx ei()P(’ 7)P(7’ )p(d)

> - sin2 (/07)--]())P(d)p(d).

From (1.55) and (1.51) we conclude that for e C

(1.56) V()(O, , X)< 1---I p(C) sin--2 (/()--/())p(d)p(d).Choose m so that p(’)(, C)> 1/2P(C). Then

=< -1/2#(c)[-up v(.)(o, , x).

The assertion of the lemma follows from (1.56) and (1.57).Lemma 1.6. I condition (B) is ]ul/illed, > O, and

then there exists A > 0 such that/or [01 < A/n

74 S. V. Nagaev

0_e_O.i

where (n) depends only upon n and lim,(n) 0, the P,,(u) are polynomialsappearing in the [ormulation o/Theorems 2--5, c(k) and d are constants.

The Proof of Lemma 1.6 is based on the decomposition (1.3) and is carriedout quite analogously to the proof of the corresponding theorem for independentrandom variables (see. [10] page 90).

Theorems 2--5 are now proved with the help of Lemmas 1.2--1.6 in exact-ly the same way as the corresponding statements for independent randomvariables (see e.g. [1] 42, 43, 45).

2. Proof of Theorems 6 and 7

1. Let us define the operator P (z) on J in the following way:

P(z)g

and denote P(O) by P. Obviously

(.1) [[P(z)--PI[ sup t__ le’-llp(, d).dX

Furthermore

()IN

(2.2)

where [z[ < A 1 < A.Since

eA11C)p(, dB),

’()I>N

uniformly with respect to #, then

(2.3) lim IIP(z)- ll 0,0

It is not hard to show, repeating the proo of Lemma 1.1 o 6 verbatim, thator [z[ < A(2.4) P"(z) 2"(z)P(z)+P"(z)P(z),where A < A is some constant which depends on M,

Limit theorems ]or homogeneous Markov chains 73

P (z) -m" R (u, z)du, P.(z) -m" R (u, z)du,

()(p () p (z)v,, t,).,(z), p)

R(u, z) is the resolvent of the operator P(z).The function

g’’(, z, u) (P(z)--P)R(u)v? fx (R(u)v/) (e"’’--l)p(, d),

R(u) R(u, 0),

for fixed u and is analytic for [Re zl <: A and

sup Ig(1)(e, z, u)[ < 1,

where E is the region exterior to the discs bounded by I and I.. We concludeinductively that the functions

g,,(, , ) (()--)R(u)]are also analytic for IRe zl < A, and, moreover,

sup Ig() (8, z, u)l _<-- sup g(1)(8, z, u).g, z, u g, z, u

Consequently, the function

(()(()-P)(), ) =j’ ()g,,(, , )()is analytic for IRe zl < A and u E. Whence i follows tha he functions

((/o. ) ((. /o. f)..

((/(/o. ) -Z ((. /o.)

are also analytic for ll < A. and e E.One analogously proves that the function (P’(), ) is analytic in the

strip IRe 1 < A for an arbitrary . Since 12()1 > 1-o1 for I1 < A., for thesevalues of log ;t(), K()= log () is by (.4) an analytic function. Let ustake for log () its principal value which tends to 0 as -+ 0. It is not hard toconvince oneself that

(.) ’ (0) =f/()p(d), K" (0) .Without loss of generality, it can be assumed that

fx I (7)P (d7(2.6) O.

Integrals are taken in the sense of Bochner. The function R (u, z) is uniformly continuouswith respect to u e E and is consequently strongly measurable, from which follows the existenceof integrals in the sense of Bochner (see [8], page 61).

76 S. V. Nagaev

K (z) and its derivatives can be expanded into uniformly convergent powerseries:

yeze(2.7) K(z)--

(.s)

K’ (z) X=i k!

’+eK"()= ,k----0

The characteristic function for =:/(x) is expressed in the following way:

/n(t) (Pn-l(it)o, (it) ),(.9)where

[,

(z, A)--A eZf()Y(d)"

Let us denote the corresponding distribution function by W(z). Let W(z)be the distribution function corresponding to the characteristic function

pn-x (h+it)v,z(h+it)(2.10) /nh(t) (pn-l(h), =(h)

(h is a real number and [hi < A). It is easy to verify the following equality:

(2.11) Wn(x Wno(X)= (pn--l(), ()) e-hVdWna(y).

Introduce the notation Fa(x)= Wa(Aa+xBna), where Ba =n2"(h),An n’ (h).

Lemma 2.1. For Ih < AK

(.) f(x) -e(x) <

where K is a constant independent o/h and x.It is not hard to show that for [hi < A 2

C e_t2ll5,

where c is a constant independent of h. For this one can use the method bywhich Lemma 1.1 was proved.

From (2.13), applying Esseen’s theorem (see e.g. 1], page 211), we easilyobtain the assertion of the 1emma. By (2.11)

Furthermore, by (2.4)

Limit theorems [or homogeneous Markov chains 77

(e.) (P-()v, ()) ’-()(P()v, ())+o(A)uniformly for Ihl < A. Moreover, it is not hard to show that

(.) (p(), ()) = +o().

Therefore,

(2.17) 1--F,[

Further reasoning essentially coincides with the reasoning of Cramdr [4].Denote K’ (h) andK" (h), respectively, by m (h) and a2 (h). Consider the equation

(2.s) m().For z sufficiently small in absolute value, this equation has a solution h whichcan be expanded in powers of :

623h -- + ....2 6

(2.19)

Evidently

Substituting h by its expansion (2.19), we obtain

Noting that 2= m2(h)/ff2, we see that

(2.21)2(2

hm(h)+K(h)

where #() is expanded into the power series #()= Co+qZ+c.,+ .:which converges for z with sufficiently small modulus.

By the lemma proved above

(.) F,(x) q,(x) +Q,(),

where Q,(x) is a function, of bounded variation and

K(2.23) IQnh (X)l < ,V/7Thus for h > O,

fo e-Zv’-VdFn, (y)1 e_ho.(a)V-v_(1/9.)VdY_Qnh(O

/2r o

if only lim inf,,_,oo’V/- > O.

78 S. V. Nagaev

Substituting the obtained expression in (2.17), we obtain

1--1n(mh) )_ (,/2(h(2.25)1-- (m(h)-a%/n)

exp In \ 2a

Now let x be a real number such that x > 1 and x--o(V’). Consider theequation

O’

which after the substitution z--x//’ becomes equation (2.18).Consequently, for sufficiently large n, equation (2.26) has one and only

one positive root h which tends to 0 as n tends to infinity.According to (2.21)

hm(h)+K(h) #

Since h z/a x/av%, then lim infooh/ > 0.Consequently, we can set in (2.25) h equal to the root of equation (2.26).

As a result, we obtain the first assertion of Theorem 1. The second assertion isproved analogously.

2. Let /,(z) be the generating function of moments of the randomvariable Sn ,’+l/(x,). Then for z in the strip IRe zi < A,

/nr(Z)--- n(k)gz(a(n+l)+kk)

Obviously /,,(z) is analytic in this strip. Consequently (ICI < A),

h C+i(:o’]h)n(k) i d C--,(Tr]k)/n2r(Z)--z(a(n+l)+kk) Z.

Setting (a(n+l)+kh)/a/ x x and x// , we obtain

h C+i(zr/k)Let z0 be a solution of the equation

d

dz[K (z) --azz] K’ (z) --.

Set C z0 and choose e < 1/2A. so that for [t] < e, [z0[ < e,

K"(Zo) >g, I, (Zo+/t)l < [ (Zo)l

For the sake of definiteness, suppose that z > 0. Obviously

Limit theorems ]or homogeneous Markov chains 79

(2.30)

where

h()

By (2.4),

It is not hard to see that

whereK(z) 1

p(z) I(t)l < max[K" (z)J/2’ lal <(Az/2) IK" (z0)l 2

Consequently

(2 34) .-<o) i n()-)"

Furthermore

(2.3z)

where

(PI(Z), (z)) (Pl(Zo), 7(Zo))-+-fl(z)(Z--Zo),

lfl (z) ___--< maxIzl<A.

d(P ()v,,

From (2.29) and (2.35) it follows that

(2.36)

In this way

o-iV;

(2.37) Ix /nK" (Zo)(Px (Zo)% (Zo) 1 +0

According to (2.29)

80 S. V. Nagaev

K"(Zo) }(2"s)’-.--"1I,1 < el2 (Zo) [" exp -nazo- n4 z

mx 1((), =())1+o(0-"’).lzl-<A

By formula (2.4) and because of uniform continuity of P(z), it is possibleto find constants c and such that for ,n > no

(2.39)

for z lying in the rectangle IRe z[ < c,

Consequently,

From (2.20) it follows that

32

(e.) K(o)--*o *" (*).2

According to (2.8), (2.19)and (2.35)

(z.4e) K"(o) o+0(,), (P(o)V,(Zo)) +0(,).

The assertion of the theorem follows from (2.30), (2.37), (2.38), (2.40), (2.41),and (2.42).

Received by the editorsMarch 18, 1959

REFERENCES

[1] B. V. GNEDENKO and A. N. KOLMOGOROV, Limit distributions /or a sum o/ independentrandom variables, Addison-Wesley, 1954.

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Limit theorems 1or homogeneous Markov chains

MORE EXACT STATEMENTS OF LIMIT THEOREMS FOR HOMOGENEOUSMARKOV CHAINS

S. V. NA(AE,V (TASHKENT)

(Summary)

This paper contains several theorems defining more precisely the convergence of homo-geneous Markov chains with an abstract state space to the Gaussian distribution. Moreover,limit theorems for large deviations are proved. The proofs are analogous to the ones in Ref. [6].