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Central limit theorems for weightedsums of exchangeable randomelements in banach spacesPeter Z. DafferPublished online: 03 Apr 2007.

To cite this article: Peter Z. Daffer (1984) Central limit theorems for weighted sums ofexchangeable random elements in banach spaces , Stochastic Analysis and Applications, 2:3,229-244

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STOCHASTIC ANALYSIS AND APPLICATIONS, 2 ( 3 ) , 229-244 (1984)

CENTRAL LIMIT THEOREMS FOR WEIGHTED SUMS OF

EXCHANGEABLE RANDOM ELEMENTS I N BANACH SPACES(')

P e t e r Z . Daf fe r

Department o f Mathematics

Vanderb i l t U n i v e r s i t y

N a s h v i l l e , Tennessee 37235

ABSTRACT

A g e n e r a l i z a t i o n o f a c e n t r a l l i m i t theorem f o r m a r t i n g a l e d i f f e r e n c e a r r a y s due t o D . L . McLeish i s o b t a i n e d f o r random e l e - ments i n a s e p a r a b l e Banach space E . T h i s r e s u l t , wi th a t echnique o f N . C . Weber, i s used t o o b t a i n a weak convergence theorem f o r weighted sums a X o f a row-wise exchangeable a r r a y (Xnk) of random elements !nn! Rkanach space E which i s un i formly 2-smooth. C o r o l l a r i e s i n c l u d e a c e n t r a l l i m i t theorem f o r weighted sums xkankXk o f an exchangeable sequence ( X k ) , and s e v e r a l weak laws of l a r g e numbers f o r such weighted sums.

0 . INTRODUCTION

In s e c t i o n 2 a n a t u r a l e x t e n s i o n o f a m a r t i n g a l e c e n t r a l l i m i t

theorem due t o D . L . McLeish [6] t o a r r a y s o f random e lements t a k -

ing v a l u e s i n a s e p a r a b l e Banach space E i s o b t a i n e d . The random

elements i n each row form a f i n i t e sequence o f m a r t i n g a l e

d i f f e r e n c e s i n E . The a d d i t i o n a l h y p o t h e s i s t h a t i s needed i n t h e

i n f i n i t e dimensional c a s e t u r n s o u t t o be t i g h t n e s s o f t h e

sequence o f row sums.

(1) Research suppor ted by a g r a n t from t h e V a n d e r b i l t U n i v e r s i t y

Research Counci l .

Copyright O 1984 by Marcel Dekker, Inc.

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2 30 DAFFER

I n s e c t i o n 2 t h e m a r t i n g a l e t echnique o f N . C . Weber [ l o ] i s

used t o o b t a i n a c e n t r a l l i m i t theorem f o r weighted row sums

Zkankxnk of an a r r a y (Xnk) of random elements i n E , i n which each

row forms an exchangeable s e t . To o b t a i n convergence t o a nonde-

g e n e r a t e (Gaussian) l i m i t , t h e h y p o t h e s i s t h a t t h e Banach space E

be uniformly 2-smooth i s used. C e n t r a l l i m i t theorems a r e ob ta ined

f o r weighted s u m s x a X of an i n f i n i t e exchangeable sequence k nk k

(Xk) i n E a s c o r o l l a r i e s . A s f u r t h e r c o r o l l a r i e s , s e v e r a l weak

laws o f l a r g e numbers f o r weighted sums z k a n k x k a r e ob ta ined

under s t r o n g e r c o n d i t i o n s on t h e a r r a y (ank) o f weigh ts .

1. NOTATION AND PRELIMINARIES

A l l Banach spaces i n t h i s paper a r e r e a l and s e p a r a b l e . A

p r o b a b i l i t y measure p i n a Banach space E i s g iven on t h e

a - a l g e b r a & o f Bore1 s e t s i n E. Sequences o f random elements

( r . e . ' s ) (X ) i n E a r e def ined on a ( s u f f i c i e n t l y r i c h ) p r o b a b i l i t y k space (n, F, P) . Expec ta t ions and c o n d i t i o n a l e x p e c t a t i o n s a r e

t aken i n t h e Bochner s e n s e ; f o r an e x c e l l e n t account of t h e i r

p r o p e r t i e s , used f r e e l y below, s e e S c a l o r a [ 7 ] .

We w r i t e E* f o r t h e dua l o f E . The c h a r a c t e r i s t i c func-

t i o n a l cp : E* -*C ( = {complex nos.)) o f a p r o b a b i l i t y measure p on E i s g iven by (P ( f ) = SEexpi f (x) p(dx) , f € E*, and determines

t h e measure ,IJ unique ly ( see , e . g . , Vakhania [ 9 ] , 54.1 f o r

p r o p e r t i e s ) . The covar iance o p e r a t o r R : E*-+E o f a weakly second o r d e r

2 p r o b a b i l i t y measure p on E ( i . e . , ,fEf (x)p(dx) < w , 'df E E*) is

t h e bounded, l i n e a r , symmetric and non-negat ive o p e r a t o r d e t e r -

mined by t h e b i l i n e a r form .fEf ( ~ ) ~ ( x ) p ( d x ) by posing

( Rf, g > = s E f ( ~ ) g ( x ) ~ ( d x ) , f , g e E*. I f p i s t h e d i s t r i b u t i o n 2 o f a s t r o n g l y s e c o n d ~ r d e r r , e . X i n E ( i . e . EIIX]~ .5 GW ), a s i s

always t h e c a s e below, t h e n R : E*--+ E i s compact. The c h a r a c t e r -

i s t i c f u n c t i o n a l o f a Gaussian d i s t r i b u t i o n o f E h a s t h e form

exp -; ( ~ f , f ) , f E E*.

For each n = 1, 2 . . . , l e t C z, C . . . . C Fn, kn C (

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CENTRAL LIMIT THEOREMS 231

be r - a l g e b r a s . An a r r a y (Xnk, ) , k = 1, ..., kn, o f r . e . ' s i n n , k

E , where Xnk i s ( c , k -& ) measurable and E(Xnk ~ j r ; , ~ - ~ ) = 0,

almost s u r e l y ( a . s . ) f o r each k = 1, ..., kn, i s c a l l e d a mar t in -

g a l e d i f f e r e n c e a r r a y i n E .

R denotes t h e r e a l l i n e , N = { I , 2 , ...) t h e n a t u r a l numbers,

and 1(A) t h e i n d i c a t o r f u n c t i o n of t h e s e t A : I(A) (a) = 1 i f

w c A, I (A) (a) = 0, i f w 4 A . The argument ~3 i s u s u a l l y SUP-

p ressed . Convergence i n p r o b a b i l i t y i s i n d i c a t e d by '$' o r

"p - l i m " . For r . e . I s (X,), X i n E wi th r e s p e c t i v e d i s t r i b u t i o n s W W

( p n ) , p , Xn+X o r ~ ~ d ~ i n d i c a t e s weak convergence of t h e measures

(i. e . , Ef (Xn) -4 Ef (X) f o r every bounded cont inuous f : E + R ) . For

a sequence o f c o n s t a n t s ( o r random v a r i a b l e s ) (c,), Xn = o (cn) P

means c - l x n ~ O . An i n d i c a t e d "k" i n "max k 'I, ltXktt i s always t a k e n

from 1 t o kn.

2. A CENTRAL LIMIT THEOREM FOR MARTINGALE DIFFERENCES

The fo l lowing r e s u l t due t o McLeish [6] w i l l be needed.

Theorem 2.1

Let (Xnk, q n , k ) , k = 1, . . ., kn, n E N , be an a r r a y of r e a l - 2

va lued m a r t i n g a l e d i f f e r e n c e s s a t i s f y i n g E [xnkl < 00 , f o r a l l n ,k .

Suppose t h a t

( i ) m p 1xnkl-%;

( i i ) s ~ p E {np jxnkl 2 , < m i

( i i i ) 5 x ~ ~ ~ - & ~ , cr2 k 0 non random.

2 Then Sn = Zkxnkd~ (0, u ) , a s n+ co . Reca l l t h a t a sequence (Xn) o f r . e . ' s i n a Banach space E is

s a i d t o be tight i f , t o every E > 0, t h e r e i s a compact s e t K C E

such t h a t sup P [Xn $k K] 5 c . ncN

Theorem 2.2

LET (Xnk, Fn, k), k = 1, . . . , kn, n E N , be a m a r t i n g a l e

d i f f e r e n c e a r r a y i n a Banach space E, s a t i s f y i n g :

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( i ) mtx llxnk1l 40, a s n + m ;

DAFFER

( i i i ) (S,), where Sn = Z k ~ n k , i s t i g h t ;

2 ( i v ) tkf (Xnk) --!Ozf, $f & 0, non-random, f o r each f r E*.

W Then S n - + Z , where Z i s a cen te red Gaussian r . e . i n E . The char -

a c t e r i s t i c f u n c t i o n a l of Z i s cp(f) = exp -: ( R f , f > , f E E*,

where R : E * A E i s t h e covar iance o p e r a t o r o f Z .

Proof

Let f E E*. Then ( f (Xnk) , < , k ) , k = 1, . . . , kn, i s a mar t in -

g a l e d i f f e r e n c e a r r a y i n R , and ~ h k o r e m 2.1 y i e l d s f (Sn) 2

= I k f ( x n k ) 4 ~ ( ~ , f ) .

This means t h a t a l l weak l i m i t s o f subsequences o f (Sn) a r e 2 t h e same: t h e p r o b a b i l i t y measure wi th marg ina l s N(0, (r f ) (use

B i l l i n g s l e y [ I ] , Theorem 5.1, and t h e c o n t i n u i t y of f E E*). Now

by ( i i i ) t h e sequence (Sn) i s t i g h t , hence by Prokhorov's Theorem

( B i l l i n g s l e y [ I ] ) c o n d i t i o n a l l y compact, and s o every subsequence

of (S ) c o n t a i n s a f u r t h e r subsequence which converges weakly t o n t h i s p r o b a b i l i t y measure. By B i l l i n g s l e y [ I ] , Theorem 2 . 3 , t h e

whole sequence (Sn) converges weakly t o t h i s p r o b a b i l i t y measure,

which i s Gaussian by d e f i n i t i o n (Kuo[S], p . 153) . Let Z denote a

r . e . with t h i s Gaussian measure. I t i s e a s i l y shown t h a t f + r L f i s a q u a d r a t i c f u n c t i o n a l d e r i v e d from t h e bounded b i l i n e a r form

( f , g)-+E(f (Z)g(Z)) on E* X E X , which g e n e r a t e s t h e covar iance

o p e r a t o r R : E*+E o f Z , d e f i n e d by ( ~ f , g) = E(F(Z)g(Z)) ,

f , g E E* ( c f . Vakhania [ 9 ] , 54 .3 .3) . I t fo l lows t h a t t h e charac-

t e r i s t i c f u n c t i o n a l o f Z i s ~ ( f ) = exp - & ( R E , f ), f E E*.q.e.d.

Remark 2.1

Condit ion ( i ) o f Theorem 2 . 3 i s e q u i v a l e n t t o and could be 2 P r e p l a c e d by t h e weak Lindeberg c o n d i t i o n IIX ) I I ( lJXnklJ 7 E)+ 0, 2k nk

f o r each c 0. I f t h e sequence (Zk[(Xnk(( )neN is uniformly i n t e -

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CENTRAL LIMIT THEOREMS 233

g r a b l e t h e n ( i ) i m p l i e s t h e u s u a l Lindeberg c o n d i t i o n 2 ZkE llXnk]J I (I/Xnkll > c ) --+ 0 , which i n t u r n i m p l i e s c o n d i t i o n ( i i ) o f

Theorem 2 . 2 . Condi t ion ( i ) i s implied by t h e c l a s s i c a l Khinchin

c o n d i t i o n f o r a Gaussian l i m i t : k l p [//xnkll > E ] - 0, a s n -r m , f o r each c 1 0.

I f t h e Banach space E p o s s e s s e s a Schauder b a s i s ( e . ) . and 1 1 c N '

i f x = t z l a i e i , x E E , t h e n Pmx =Em a . e and gx = (1-P ) x = i=l 1 1 m

~ ~ m + l a i e i d e f i n e s , f o r m e N , t h e p a r t i a l sum and r e s i d u a l oper-

a t o r s on E . I t f o l l o w s from t h e proof o f Lemma 4, p . 259, of Dun-

f o r d and Schwartz [2] t h a t a s e t K C E i s c o n d i t i o n a l l y compact i f

and o n l y i f

( i ) supxcKllxll < m and

( i i ) t o every c > O t h e r e i s m E N such t h a t supxeK E 11%;1l~ .

Lemma 2 . 3

Let (X ) be a sequence o f r . e . ' s i n a Banach space E with a n Schauder b a s i s , and s a t i s f y i n g l i m supn~l lxJ / '1 = 0,

a+ 00 [ I l X J > Oc I 1 p <m. Then (Xn) is t i g h t i f and o n l y i f l i m r u p n ~ l l e ~ ~ ~ P = 0.

in+ w

Proof -- Let (X,) be t i g h t , and E > 0 given . Choose a such t h a t P

llXnll I [ 1/xnl[ > a ] < I, f o r a l l n. Next, f i n d K compact, such t h a t

PIXn 4 K] 4 I, f o r a l l n . Then E l1xn/l '1 [Xn+Kl g $P [xn& K] + 2 aP

E 1 1 ~ ~ 1 ~ 1 f + 5 = L . NOW s i n c e K i s compact, we have m-+ l i m m

P S ~ P ~ E I / Q ~ X ~ I I ~ = ~ ~ ~ s ~ P ~ E ~ / \ X ~ I [ ~ ~ ~ ~ ~ I I L C ~ U P ~ E I I X J ~ ~ I [ X ~ + ~ ( ] < Cr,

where C = supnllQmllP. S i n c e r i s a r b i t r a r y , m- l i m m S U ~ ~ E / / P X J I ~ = 0 .

Conversely, suppose ~ ~ m m s u p n E ~ I ~ X ~ / = 0. Let & > 0 be g iven .

To every k € N , t h e r e i s mk 6 N such t h a t EQ,, xn/lP < k-12-k-l k

c , f o r a l l n , and m & m k . By P [ I I x , I I ~ > ~ I ~ E

< ~ - ' E I I X ~ I I ~ ~ lixnll > a1 f i n d a such t h a t P [ ~ ~ x ~ I J ~ > a] < -, f o r a l l n. 2 1

Now l e t A. = (x € E : IIx\IP L a ) - and Ak = (X € E : l lekxl lP g k- ) , 00

k r N . Put K = nkSOAk. I t is e a s i l y checked t h a t K i s condi- * C

t i o n a l l y compact. Now P [Xn $ K] = P [Xn G U k = O ~ k ] =

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234 DAFFER

3. LIMIT THEOREMS FOR WEIGHTED SUMS OF EXCHANGEABLE RANDOM ELEMENTS

We begin by recalling the concept of exchangeability.

Definition 3.1

A sequence (Xn) of r.e.'s in the Banach space E is said to be

exchangeable if, to each fixed n e N, the joint distribution of

(Xql, ..., Xgn) is the same as that of (XI, ..., Xn) where (~1, ..., 7m) is an arbitrary permutation of the integers

(Iy ...y n).

Clearly, exchangeable T.e.'s are identically distributed.

Definition 3.2

A Banach space E is said to be uniformly p-smooth, 1 L_ p 2, if ?(t) = 0(tP) as t 0, where the modulus of smoothness ~ ( t ) is

defined by ~ ( t ) = sup(:(lx + y l l + qllx - yII - 1 : llxll = 1,

l l ~ l l = t) . The spaces lP and L~ are uniformly 2-smooth for 2 g p <a,

and uniformly p-smooth for 1 < p L - 2.

The specific property of uniformly p-smooth spaces that will

be needed below is the following: If E is isomorphic to a uni- formly p-smooth Banach space, then E 4 C ~ ~ = ~ E I I X ~ I ~ for every martingale difference sequence XI, . . . , Xn with E l l ~ ~ l l ~ ~ m (Hoffman-JBrgensen and Pisier [3]) .

Given an array (X ), k = 1, ..., kn, of r.e.'s in E, define nk the o-algebras (9 ), k = 0, 1, . . . , kn, n € N, as follows:

n,k

tkn X .), k = 0, 1, ..., k - 1, and <,k=g(Xnl' "" 'nk9 j=k+l nl n 7mJkn = Q(Xnl, . . . , X ) . For a r.e. X, we abbreviate

nkn

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CENTRAL LIMIT THEOREMS 235

Theorem 3 .1

Let (Xnk), k = 1, ..., kn, be an a r r a y of row-wise exchange-

a b l e random elements i n a uniformly 2-smooth Banach space E . Let

= 1, ..., kn, be an a r r a y o f r e a l c o n s t a n t s , Suppose: 2

(llXnlll )ncN i s bounded s t o c h a s t i c a l l y by a random v a r i a b l e 2

X, f o r which ElXi 4 W ; - 1

maxkllxnkl l = op ( l a n k / ) ) ; 2 s"~nxkank' ;

2 2 t k a i k f 2 ( x n k ) i b f , Of 2 0 non-random, f o r each f c E* ;

(Xnl)nsN i s a t i g h t sequence.

(X ))ZZ, a cen t e r ed Gaussian random Then Zkank('nk - En, k - l nk element i n E .

Proof

(X 1 ) . Then (Ynk), k = 1, ..., Let 'nk = ank(Xnk - En, k-1 nk kn'

i s a ma r t i nga l e d i f f e r e n c e a r r a y i n E. We v e r i f y t h e hypotheses of

Theorem 2 . 2 f o r (Ynk) . For ( i ) we have

~ ~ l l ~ n k l l

f lank\ IIXnkll + mtx lankl (IE,, k-1 (Xnk)ll

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by (1) and (3) .

For ( i v ) , w r i t e

Now

A s f o r t h e o t h e r term, we have

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2 2 4 [ F a n k f ( x n k ~ ~ 'I2 [%ankf * (Xnk) 11 I/z -O p

by (5) and what was j u s t proved. Hence

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CENTRAL LIMIT THEOREMS

non-random, by (5).

Finally we need to show (iii) of Theorem 2.2, that

(t Y ) is tight. To accomplish this, let h : E+C[O,l] be an k nk isometric isomorphism embedding E into the universal space

Z

C [O, 11 . Then E = h(E) c C [O, 11 is a Banach space, subspace of

C[0,1], and is uniformly 2-smooth. Now h(Y ) = a (2 - .. nk nk nk (2 ) ) , where Xnk = h(Xnk) , and (h(Ynk)) , k = 1, . . . , k is En,k-l nk ,+

a martingale difference array in E. Then

.., (linearity of Qm and uniform 2-smoothness of E)

Now since (X ) is tight in E, (cl) is tight in C [0, l] . From (1) nl

] = 0'

this

and Lemma 2.3 with p = 2, then yields 2

i m E = 0. m+ca

2 This with (3), then yields lim = 0. But

m-+ w implies

lim sup~~l~J\~ll = 0, m-+m n k

and ( T ~ < ~ ) ~ ~ ~ is uniformly integrable. (Indeed,

S U P E I I ~ < ~ U n

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DAFFER

impl ies uniform i n t e g r a b i l i t y ) . Thus Lemma 2.3, appl ied t o ,. ( Z , G ~ , ) ~ ~ , , t h i s t ime f o r p = 1, y i e l d s t h a t (JlkYnk)nN is t i g h t

i n C[0,1] . Hence (fkYnk)neN i s t i g h t i n E . Theorem 2.2 now y i e l d s W

(X ))-+Z, a cen te red Gaussian r . e . i n kYnk = z k a n k C X n k - En, k-1 nk

E . q.e.d.

Remark 3.1

Theorem 3.2 could be genera l ized by no t r e q u i r i n g ( 3 ) - - t h a t

( ~ ~ a : ~ ) be bounded--but a t t h e expense o f in t roduc ing some 2

r a t h e r unpleasant hypotheses. Write An = xlkank. For i n s t ance , i f

( 6 ) i s changed t o read Ef (Xnl)f (Xn2) = O ( A ~ ) ; and i f t h e fo l lowing 2 - 1

two hypotheses a r e in t roduced; f i r s t : E I ( X , ~ J I = O(An ) , and

second: f o r an isometry h : E--.C[O,l] we have

limm_msupnAnE~~~h(~n1)I~2 = 0 ; then t h e conc lus ion of Theorem 3.1

w i l l fol low, mu ta t i s mutandis, without (3). Indeed, i f it i s

assumed t h a t l i m m _ _supnZk (ankl P ~ l l ~ m h ( ~ n l ) ~ I P = 0 , 1 5 p < 2, t h e

requirement o f uniform 2-smoothness can be dropped t o t h a t of un i -

form p-smoothness, and t h e conc lus ion of t h e theorem w i l l fo l low.

Here, however, we must have l i m 1 la l P = Cb i f t h e l i m i t i s t o - be non-degenerate; s u p n r k l ankl < , 1 4 p ( 2 , f o r ce s

l i m n + _ x t a i k = 0, which e n t a i l s convergence i n p r o b a b i l i t y of 2 2

(X ) ) t o t h e zero element ( p [Xank f (Xnk) Ekank('nk - En, k-1 nk

> CI 4 C ' A , I I ~ ~ ~ ~ E 1x1 2- 0).

Remark 3.2

sup A = supn zka:k < m can be deduced from (5) and t h e n n

fol lowing hypothes i s : There i s an c > 0 and f E E* such t h a t

l iminf n-co P [ f ( X n l ) > c ] > 0. (See Lemma 3.2 below.) Thus, f o r a l l

s u f f i c i e n t l y l a r g e n , p o s i t i v e p r o b a b i l i t y f o r Xnl l i e s i n a h a l f -

space an E removed from t h e o r i g i n .

Remark 3 .3

Note t h a t (4) o f Theorem 3 .1 i s implied by maxk la nkl = o(log-1'2kn).

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CENTRAL LIMIT THEOREMS

Remark 3.4

I t i s no t d i f f i c u l t t o d e r i v e from ( I ) , (3) and (5) o f 2 2

Theorem 3 .1 t h a t t h e sequence ( z k a n k f (Xnk)) ncN i s un i formly

i n t e g r a b l e f o r each f E EX, and hence (5) could be r e p l a c e d by

a p p a r e n t l y s t r o n g e r c o n d i t i o n l i m 2

9'" E l ~ ~ a ~ ~ f ~ ( ~ ~ ~ ) - c r ~ 1 = 0, which i n t u r n i m p l i e s l i m AnEf (Xnl) = o f E E*.

n-m E'

239

t h e

The fo l lowing easy lemma w i l l be needed. Note t h a t it f a i l s

i f X i s no t non-negat ive. 1

Lemma 3.2

Let (X ) be i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s wi th n X1 $ 0 and X L. 0, a s s . , and (ank) , k = I , . . .k be non-negat ive

1 - n P numbers. I f ~ : : l a n k ~ k - c , a c o n s t a n t , t h e n sup x k a n k < m .

Proof - C C

Let Sn = x kankxk and Sn = 'nl [ s n 4 c + l ] Then P[Sn # S,] 3

0. Let ~ > O a n d 6 > 0 b e s u c h t h a t P [ X 1 > e ] > S . T h e n l + c 2

sC:,Z a E I I n - k nk [Sn<c+l] [Xk> E ] '

Now P[Sn c + 1 and Xk> c ] >

6 /2 f o r a l l s u f f i c i e n t l y l a r g e n and s o , t a k i n g e x p e c t a t i o n s , -1 -1

c 8 / 2 Z k a n k L_ 1 + c , o r r k a n k 4 E 6 (1 + c) f o r a l l s u f f i c i e n t l y

l a r g e n . q .e .d .

I t fo l lows d i r e c t l y from D e f i n i t i o n 3 . 1 t h a t E n , k - l (x nk I =

1 zk2 X . k = 1, . . . , kn, i f (Xn) i s an exchangeable k - k + 1 J k n J ' n sequence ( c f . Kingman [4] ) .

A sequence (Xn) o f r . e . ' s i n a Banach space E i s s a i d t o be

bounded s t o c h a s t i c a l l y by a random v a r i a b l e X ( i n R) i f

P[llXnll 1 X] P[X 1 X I , f o r every x 1 - 0 . ( C l e a r l y X 3. - 0 , a lmost

s u r e l y ) .

For a sequence (X ) o f r . e . I s l e t = n

kn X I > x k , j k + l x j , k = 0, 1. . . - , kn -1, <,kn =

6(X1, . . > Xkn) .

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DAFFER

Corollary 3.3

Let (X ) be a sequence of exchangeable random elements taking n values in a uniformly 2-smooth Banach space E, and satisfying (i)

~ 1 1 ~ ~ 1 1 ~ ~ m and (ii,) Ef (X )f (X2) = 0, for every f c E*. Let 1 (ank), k = 1, ..., kn, be an array of real constants satisfying:

7 L

(a) lim ank = 0; n+m2k kn - k + 1

- 1 (b) maxkllxkl = op((max lank/ ) ;

2 2 (c) xkankf (xt)Pd, 2 ). 0 non-random, for every f e E* f f -

(X ))2~, a centered Gaussian random Then Zkank('k - k element in E.

Proof

(I), (2), (4) , ( S ) , (6) and (7) of Theorem 3.1 are trivially

satisfied. If X = 0, a.s., the result is trivial. If 1 PIX1 # 0] > 0, then for some c z0 and 6 > O and some f € E* we have

L P[f (XI) > E ] > 6. Applying Lemma 3.2 to (f (X,)) , (c) then yields 2

supnxkank<m, which is (3) of Theorem 3.1. The results follows.

Remark 3.5

Referring to remark 3.4, it is seen that the hypotheses of 2 2 Corollary 3.3 imply that Ef (X )lim A = c f , SO that

2 2 1 n 4 w n Qf = const. Ef (XI), f E E*.

Remark 3.6

We do not see how to obtain a nondegenerate limit result for

the case of uniform p-smoothness, 1 f p < 2. The only place in the proof of Theorem 3.1 where uniform

2-smoothness is needed is in obtaining the tightness of the

sequence ( tkYnk)neN- If it is replaced with uniform p-smoothness

for 1 L - p < 2, and a stronger condition is placed on the array (ank) of weights, the result is a weak law of large numbers. We

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CENTRAL L I M I T THEOREMS

s t a t e t h i s f o r t h e s p e c i a l c a se of a sequence (X,) o f exchange-

a b l e r . e . ! s .

Co ro l l a ry 3.4

Let (X ) be a sequence o f exchangeable r . e . ' s i n a uniformly k 2 p-smooth Banach space E, with 1 4 p < 2 , s a t i s f y i n g E / I x ~ I < . Let ( a ), k = 1, ..., k be an a r r a y of r e a l cons t an t s s a t i s f y i n g nk (b) of Co ro l l a ry 3 .3 and (d) supn xk lank 1 P< m . Then

Proof

A s before , cons ider t h e mar t inga le d i f f e r e n c e a r r a y (Ynk)

where Y nk = a nk(Xk - En,k-l(Xk)). Af t e r embedding E i s o m e t r i c a l l y

i n t o C[0,1] , a s i n t h e proof of Theorem 3.1, we have

E1~s~k<k~~P & zk tankI PEl~s'nk - sEn, k- ('nk)ll 4

c 2 P - 1 ~ k lank I * [ E I I % ' ~ ~ I I ~ + E1~sEn,k-l ('nk)(I '1 4

2 1 a E + 0 a s m , us ing (d) . As i n t h e proof of

Theorem 3.1, (t 7 ) i s uniformly i n t e g r a b l e , and an a p p l i c a t i o n k nk

of Lemma 2.3, f o r p = 1, y i e l d s t i g h t n e s s of (x Y ) . k nk 2 We need t o show t h a t (c ) o f Co ro l l a ry 3.3. ho ld s with cf = 0,

f o r each f E E*, and t h a t both ( i i ) and (a ) o f Co ro l l a ry 3 .3 a r e

unnecessary. We f i r s t no t e t h a t i f X1 = 0, a . s . , t h e r e s u l t i s

t r i v i a l , and we assume PIX1 # 01 > 0. But t h i s t oge the r with (b)

o f Co ro l l a ry 3.3 e a s i l y impl ies t h a t limn+oomaxklankl = 0. Now,

2 f o r e > 0 and f E E*, P [ t k f (Ynk) > E] ( E ~ ~ ~ E ~ ~ Y ~ ~ ) 4

2 2 2 i1 c 2 &I j2 X ~ ~ ; ~ [ E I I X ~ I I + ~ p ~ , ~ - ~ (xk)ll I 6 2 2

4 g1 f EllXll/ ,Yka;X g const.maxklankl 2-P,T+a:k4 O , by ( d ) , s i n c e

2 max / a [ 2 p - ~ , a s n . ~ h u s t ~ f C ( , ~ ) ~ O a s n - + m , f o r every

k nk

f E E X , and hypotheses ( i i ) and (a ) o f Co ro l l a ry 3 .3 were not

needed t o a r r i v e a t t h i s . Checking t h i s proof o f Theorem 3.1, it

i s seen t h a t t h e only p l ace t h a t (a) and ( i i ) o f Co ro l l a ry 3 .3 were

needed was i n v e r i f y i n g ( i v ) o f Theorem 2.2. But t h i s has been

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242 DAFFER

v e r i f i e d above without t hese condit ions. Corollary 3 . 3 now y ie lds

the conclusion. q.e.d.

By s t rengthening t h e hypotheses on t h e a r r ay (a ), t h e cen- nk t e r i n g a t condi t ional expectat ions can be el iminated.

Corollary 3.5

Let (X ) be a sequence of exchangeable r . e . ' s i n a uniformly n 2 2-smooth Banach space E , such t h a t E(!x# < 00 . Let (a ), k = nk

1, ..., kn, be an a r r ay of r e a l cons tants s a t i s f y i n g (b) of

Corollary 3 . 3 and

(el c l a n k t -. 0 a s n + rn

m e n Sn = t k a n k x k 2 ~ , a s n--t m.

The freedom of choice of t h e r - a lgeb ras Cmk can be exploi ted

t o weaken t h e hypotheses on t h e a r r ay (ank).

Theorem 3 . 6

Let (X ) be a sequence of exchangeable r . e . ' s i n a uniformly n 2 2-smooth Banach space E , and s a t i s f y i n g ( i ) ~11x41 ( 00 and

( i i ) Ef(Xl)f(X2) = 0, f o r each f c E*. Let k 1 f o r a l l n, and n n

let (a ), k = 1, ..., 1 be an a r r ay of r e a l constants. Let (b) nk n

and (c) o f Corollary 3 . 3 be s a t i s f i e d together with

- 1 ( f ) ,,l2m&knln = 0

(X - E n , k - l ( ~ k ) ) ~ ~ , where Z i s a centered Gaussian Then x::lank k r . e . i n E.

Proof

The proof follows t h a t of Corollary 3 . 3 . (a) of t h a t

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CENTRAL LIMIT THEOREMS 243

Coro l l a ry i s (3) o f Theorem 3.1, which was used s o l e l y t o v e r i f y

( i v ) o f Theorem 2.2. But ( i v ) can be v e r i f i e d u s ing ( f ) a s

fo l lows (Y - nk - ank(Xk - En, k-1 (xk) 1) :

L - c mex a Z [ I + log ln - nk

I f PIX1 = 01 = 1, t h e theorem i s t r i v i a l . I f we assume t h a t 2 PIX1 # 0] 7 0, then it fo l lows from Lemma 3.2 t h a t suPnxkank< 03,

from which a f o r t i o r i sup max a 2 < Oo. Now ( f ) y i e l d s n k nk 2 2

l i m n + & l t k a n k f (En, k-l (Xk)) I = 0 , from whish

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244 DAFFER

2 2 (X ) ) = 0. As in the proof of Theorem 3.1, p - limn t a n k f (En,k-l k

we obtain p - lim 2 (X ) ) = 0. Hence (iv) of n zkankf ('kIf ('n,k-1 k Theorem 2.8 follows, completing the proof. q.e.d.

Remark 3.7

Some applications call for treating the weights (ank) them-

selves as random variables (Taylor [$], Chapter VI). This poses no

problem if the weights (a ) are independent of the random elements nk

( X ) Hypotheses such as (3) of Theorem 3.1 are replaced by

expressions like sup$ ~a~ < M , involving expectations. k nk

REFERENCES

[ I ] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.

[2] Dunford, N. and Schwartz, J. (1958). Linear Operators I. Interscience, New York.

[3] Hoffmann-~drgensen, J. and Pisier, G. (1976). The law of large numbers and the central limit theorem in Banach spaces. Annals of Prob. 4, 587-599.

[4] Kingman, J.F.C. (1978). Uses of exchangeability. Annals of Prob. 6, 183-197. - -

[5] Kuo, H. H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics Vol. 463. Springer-verlag, Berlin.

[6] McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Annals of Prob. 2, 620-628.

[7] Scalora, F. S. (1961). Abstract martingale convergence theorems. Pacific Jour. Math. 2, 347-374.

[8] Taylor, R. L. (1978). Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces. Lecture Notes in Mathematics Vol. 672. Springer-verlag, Berlin.

[9] Vakhania, N. N. (1981). Probability Distributions on Linear Spaces. Elsevier-North Holland.

[lo] Weber, N. C. (1980). A martingale approach to central limit theorems for exchangeable random variables. Jour. of Applied Prob. 17, 662-673. - -

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