Complete convergence for weighted sums of arrays of random elements

J.Korean Math. Soc. 32 (1995), No. 4, pp. 679-688

COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF ARRAYS OF RANDOM ELEMENTS

500 HAK SUNG

1. Introduction

Let (B,1111) be a real separable Banach space. Let (0.,F,P) denote a probability space. A random elements in B is a function from n

into B which is F-measurable with respect to the Borel a-field B(B) inB. An array {Xnk} of random elements in B is said to be uniformly bounded by a random variable X if for all nand kand for each t > 0,

P(IIXnkll >t) ~ P(!Xj >t).

A separable Banach space B is said to be of typep,15 p5 2, if there exists a constant C such that

n n

EIII: Xkll

I:

k=l k=l

for all independent random elements Xl, ... , Xⁿ in B with mean zero and finite p-th moments. A sequence {Un} of random elements in B is said to converge completely to zero if for each E> 0,

I:

n=l

Note that complete convergence implies almost surely convergence by Borel Cantelli lemma.

Received August 26, 1994.

1991 AMS Subject Classification: 60F15, 60B12.

Key words: complete convergence, random elements, Banach space.

This paper was supported (in part) by NON DIRECTED RESEARCH FUND, Korea Research Foundation, 1993.

Pruitt[9] investigated the complete convergence of

00

LankXk

11:=1

when {Xn } are i.i.d. random variables, and {ank,n ~'l,k ~ l} is a Toeplitz array. Recall that an array {aRk} isa Toeplitz if

(i) limn_coank =0 for all k, and (ii) _L:~1 lankI :::;C for all n.

Generalizing a result of Pruitt, Rohatgi[lO] proved the following result.

THEOREM 1.1. (Rohatgi[lO]) Let {Xn } be asequence of indepen- dent random variables with EXn = 0 for all n which is uniformly bounded by a random variableX withE/X/1+l/T <⁰⁰ for some r > O.

Let {ank} be a Toeplitz array satisfying maxklankI= O(l/nT). Then L:~1ankXk ~0 completely.

Padgett and Taylor[7] considered the problem of extending of Ro- hatgi's theorem for real-valued random variables to Banach-valued ran- dom elements. They noted that Rohatgi's theorem can not be ex- tended directly to separable Banach spaces. Wang and Rao[16] ex- tended Rohatgi's result to the uniformly tight random elements: Tay- lor[14] obtained complete convergence for rowwise independent, uni- formly bounded random elements in B-convex space. As a corollary of this result, he obtained a version of Rohatgi's result in B-convex space.

The main theorem of Taylor isas follows.

THEOREM 1.2. (Taylor[14]) Let {Xnk} be an array of rowwise in- dependent random elements in a separable B-convex space B with EXnk = 0 for all n and k. Let {ank} be a Toeplitz array satisfying maxklankI= ^O(l/nT) for some r > O. If{Xnd is uniformly bounded by a random variable X with EIXII+1/T< 00, thenL:~1ankXnk ~ 0 completely.

The purpose of this paper is to extend Taylor's theorem to general

Banach spaces. .

The convergence of the form

(1.1)

.L

k=l

Complete convergence for weighted sums of arrays of random elements 681

can be founded in Bozorgnia, Patterson, and Taylor[2],Sung[12], Tay- lor and Hu[15], and Wang, Rao and Yang[17], where {Xnk ,^{1 ~} k ~

n, n 2:: I} is an array of rowwise independent random elements, and {ank'1 ~ k ~ n, n 2:: I} is an array of constants. It should be noted that classical limit theorems for independent random variables hold for random elements under the additional condition of convergence in probability; see [1], [3], [4], [5], [6], [11], [12], and [17]. For example, Wang, Rao, and Yang[17] showed that (1.1) holds if and only if

(1.2)

L

k=l

when ank = 1/n^l/p,1 ~ k :::; n, for some 1 :::; p < 2, and {Xnk}

is rowwise independent, uniformly bounded by a random variable X with EIXI^2p < ^00.

In this paper, we show that L:~lankXnk -+ 0 completely if and only if L:~lankXnk -+ 0 in probability under Taylor's conditions without B-convexity. We also obtain the equivalence of L:~lankXnk

-+ 0 in probability and in L1 under some restrictions on {ank} and {XnA:}.

Throughout this paper, C will always stand for a positive constant which may be different in various places.

2. Main Results

To prove theorem 2.2, we need the following lemma.

LEMMA 2.1. (de Acosta[l])LetXl,'" ,Xn beindependent random elementsin B withEIIXkllr <⁰⁰ for k= 1,'" ,n and 1 ::; r ::;2. Then

n

EIIISnll- EllSnl1 Ir:::; Cr

L

k=}

where Sn = _L:~=lXk , and Cr is a positive constant depending only on r.

The next theorem, our first main theorem, shows the equivalence of L:~lank X^nA: -+ 0 in probability and inLI under some restrictions on {and and {Xnd.

THEOREM 2.2. Let {Xnk} beanarray ofrowwise independent ran- dom elementsin B such. that maxn,kEIIXnk11a < 00 for some a > 1.

Let {and be a Toeplitz array satisfyingmaxklankl ⁼ O(l/n^r)for some r >O. Then the following statementsareequivalent.

(i) _~::1ankXnk --+ 0in probability as n --+ 00.

(ii) ~;'=1ankXnk --+ 0 inLl as n --+ 00.

Proof. Since (ii)::} (i) is obvious, we will show (i)::}(ii). Assume that _~::1ankXnk --+ 0 in probability as n --+ 00. Since _~::1 lankl is bounded, we can choose a sequence {rn} of integers such that

Then

(2.1) _Ell

f

_~ f

k=rn +l

since maxn,k EIIXnk!l is finite. Hence it is enough to show that

(2.2)

L

Let f3 ⁼ min{2,a}. From Lemma 2.1,

rn rn

El ¹¹

L

L

k=1 k=1

since f3 >1.Hence

(2.3)

lIt

k=1 k=1

Complete convergence for weighted sums of arrays of random elements 683

Note that

IlL

by (i) and (2.1). Combining this result and (2.3) gives the desired result (2.2).

The following lemma plays an essential role in our second main result.

LEMMA 2.3. (Sung[12]) Let Xl,··· ,X_n be independent random elementsin B such that

IIXI:II ::; ^bre , 1::; k ::; n, andlet Sn =

E:=l

0

n

E[exp(t11Sn11)] ::; exp{tEll Sn11+2t²

L

1:=1

The next theorem extends Rohatgi's result to separable Banach space. It is also a generalization of Taylor's theorem [14].

THEOREM 2.4. Let {Xnd be an array of rowwise independent random elements in a separable Banach space B which is uniformly bounded by a random variable X withEIXII+I/r <⁰⁰ for somer >o.

H Toep1itz array {anI:} satisfies maxk lankI⁼ O(l/n r), then the fol- lowing statements are equivalent.

(i) E~lankXnl: -

0

(ii) _L:~1ankXnk - 0 completely as n - ^00.

We need only to prove (i)=>(ii). The proof is completed by the following three lemmas, since

00

C

{L

i}

k=l

U{lIankXnkll ?

i

k}

U{lIankXnkll ? n-^a for at least two values ofk}.

The following Lemma 2.5 and Lemma 2.6 are in Taylor[14](or see Ro- hatgi[l'O]J. The proof of Lemma· 2.7 is different from that of Lemma 3 in Taylor[14]. It seems that the proof of Lemma 3 in Taylor for B-convex Banach space can not be adapted to general Banach spaces.

LEMMA 2.5. IfEIXIl+¹/r < ⁰⁰ and maxklankl = O(l/n^T), then for everye> 0

. 00

.2:

n=l

LEMMA 2.6. IfEIXIl+¹/T < ⁰⁰and maxklankl =O(l/n^r),then for a <r!(2r+2)

.2:

n=l

LEMMA 2.7. IfEIXIl+¹/r < ⁰⁰ and maxk lankl =O(l/nr ), then

.2:

k=l

implies

00 00

L

e)

n=l k=l

where 0< a < r.

Proof. Define Znk = XnkI(/lankXnkll < n-^a). Then EIIZnkl! ::;

EIXI. Let {r_n} be as in the proof of Theorem 2.2. By Markov in- equality

00 00

L

p(1I

€)

n=1 k=1

00 rn ^{00 00}

:5

2: p(1I

2: p(1I 2:

n=1 k=1 n=1 k=rn+l

00 r n 2 00 00

5

2: p(lI2:

2: 2:

n=1 k=1 n=1k=rn+l

00 ~ "~

5 L_n=1

p(1I

i)

:2·

Complete convergence for weighted sums of arrays of random elements 685

Hence it is enough to show that

(2.4)

Fix n ;::: 1. Let t = 41ogn/€. Since IIankZnkll < n-o, it follows by Markov inequality and Lemma 2.3 that

(2.5)

Now we calculate the power of exp in the last expression of (2.5). From triangular inequality

r n

Ell

E

k=l r n

=EII

E

r n r n

$EII

E

E

k=! k=!

00 00

$EII

E

E

k=l k=rn+1

rn

+Ell

E

k=!

00 EIXI rn

$EII

E

E

k=! n k=!

since the first term in the last expression converges to 0 by Theorem 2.2, the second term converges to 0 clearly, and the third term converges

to 0 by the following fact.

Tn

Ell

L

k=l

T n

~

L

00

~EIXII(lXI>CnT-a)

L

k=l

~CEIXII(IXI >CnT-a) ^-+o.

Tⁿ 1 Tⁿ EIXI ⁰⁰ EIXI

LE!lankZnkll² ~ n a LE!lankZnkll ~

--;;c;- L

k=l k=l k=l

Thus, for any fixed T/ > 0, the power of exp in the last expression of (2.5) is bounded by

log2n Cl / "

-2Iogn+T/logn+C--e ogn n n^a

for alln sufficiently large. ChooseT/ such that -2 + T/ < -1. Then

00 { log2 n }

~ exp -2 log n + T/logn + C---;:;;:-eCIOgn/n"

00

~CLexp{-21ogn+T/logn}<^00.

n=l

Therefore the desired result (2.4) holds.

In Theorem 2.4, the convergence in probability is obtained by im- posing an additional geometric condition on B.

COROLLARY 2.8. (Taylor[14]) Let {Xnk } ^and{ank} beasin Theo- rem 2.4. H EXnk = 0 and Banach spaceisB-convex, then

L

k=l

Complete convergence for weighted sumsofarrays of random elements 687

Proof. By Theorem 2.4, it suffices to show that

L

k=1

Note that B-convexity implies that B is of type pfor somep >1, see Pisier[8] (or Taylor[14]). Let f'

=

00 00

Ell

L

L

k=1 1:=1

Thus the proof is complete.

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Department of Applied Mathematics Pai Chai University

Taejon 302-735, Korea