Almost sure convergence for weighted sums of I.I.D. Random variables

ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF I.I.D. RANDOM VARIABLES

,,'..sPQ. lI,AK SUNG ,_{I " , ,', '}

1.

Introduction

Let {X, X

n 2: 1} be a sequence of independent and identically distributed (LLd. ) random variables. Let {ani, 1

n, n 2: 1} be a triangular array of constants. The purpose of this paper is to find sufficient conditions on {ani} and X such that

(1)

n

Lani(Xi - EXi )

0 a.s.

i=l

Some convergence theorems for these weighted sums have been ob- tained by Choi and Sung [2], Chow [4], Chow and Lai [5], Stout [8], Teicher [9], and Thrum [10]. For example, it has been shown in Choi and Sung[2] that conditions EIXI <

and

lanil

O(l/n) imply (1). The first major result of this paper is to generalize the result of Choi and Sung[2] for the case of random variables with finite p-th moment(l

< 2).

In the

case of ani

adbn for 1

n and n 2: 1, considerations concerning (1) can be found in Adler and Rosalsky [1]' Fernoholz and Teicher [6], Jamisons, Orey, and Pruitt [7], and Teicher [9]. Adler and Rosalsky [1] have shown that if {X, X

is a sequence of LLd. random variables with EIXIP <

for some 1

< 2, and {an} and {bn } are

Received June 29, 1995.

1991 AMS Subject Classification: 60F15.

Key words: almost sure convergence, weighted sums, strong law of large num- bers.

This paperwassupported (in part) by NON-DIRECTED RESEARCH FUND, Korea Research Foundation, 1994.

constants satisfying 0 < b

i

(2)

and

(3)

then

(4)

L

^lail

^O(b

Note that Adler and Rosalsky's theorem includes Kolmogorov strong law of large numbers(SLLN) as a special case, where

= 1 and

= n for n

1, but not Marcinkiewicz SLLN. The second major result is to improve Adler and RosalskY's theorem by showing that (i) condition (3) is unnecessary when 1 <

< 2, and (ii) condition (3) is essential whenp

It proves convenient to define 19x

max{l,logx}, where log x de- notes the natural logarithm. The symbol C denotes a constant which is not necessarily the same one in each appearance.

2 •. Main Results

The following two lemmas will be used in obtaining our first main result. The proof of Lemma 1 is similar to that of Lemma 2.2 of Choi and Sung [3].

LEMMA

1. H E1XIP <

for some 0 <

< 2, then

00

1 .

" EX

I(IXI < ) <

~ n2/p(1gn)l-2/p -

(lgn)l/

Proof

~ 1 l~

~ ^EX

^l(IXI < n )

~ n 2/ p(lgn)1-2/ p -

(lgn)l/p

== ~ n2/p(lg~)l-~/P ^"

~ ^{EX2l( (i - 1)1/P} IXI i 1 / P )

~ (lg(i -1))1/p < ~ ^{(lgi)l/P (lgO}

¹⁾

~

(i - 1)1/p i1/p

= ~

EX

l( < IXI < )

~

(lg(i - 1))l/p - (lgi)l/p

~

1

~

~

2 (i _1)1/p i1/ p i

~ ^C ~EX l((lg(i -1))1/p < IXI _~ (lgi)l/P) i2/ p(lgi)1-2/p

~

(i - 1)1/P i1/ p i

~ C~P((lg(i ^{-1))1/P < IXI} ~ (lgi)l/p\gi

~

CEIXI

<

since the first inequality follows from the following.

~

1

~n2/p(lgn)1-2/p

1

<C

- i x 2/ p(logx)1-2/p -

i2/ p(lgi)1-2/p

LEMMA

2.

<

for some

> 1, then

o

Proof. The proof is similar to that of Lemma 1 and omitted. 0 The next theorem, our first main result, is an extension of Theorem 5 of Choi and Sung [2]. They have proved Theorem 1 when

1.

THEOREM 1.

Let {X,Xn,n :;::: I} be a sequence of LLd. random variables with EIXIP <

for some 1 ::; p < 2. Let {ani, 1 ::;

n, n :;:::

I} be a triangular array of constants satisfying

max la ·1 - Q( 1 )

l:5i:5_n m -

n1/p(lgn)1-1/p'

Then (1) holds, Le.,

ani (Xi - EXi ) -- 0 a.s.

Proof Define

Then

+ X::'

X n - EXn for n :;::: 1. To prove

n

limsup L ani (Xi - EXi) ::; 0 a.s.,

n->oo i=l

it

enough to show that

(5)

and

(6)

n

lim sup L ani X : ::; 0 a.s.

n->oo i=l

n

lim "aniXr

0 a.s.

n-JoOO

L...J

i=l

From the inequality

1 +

+

^{1xl /2 for}

^R, we have for

t>O

By the independence of

n n

E[exp(t L ^{aniXDl =} IT E[exp(taniXDl

i=l i=l

Let E> 0 be given. By putting

= 2Ign/E, we obtain

n n

P(Lani X : > E) ~ e-t€E[exp(t L ^aniXDl

i=l i=l

) -tE {

t

( t ) ~ ^/2}

(7

exp C n 2/ p(lgn)2-2/p exp C Ign ~EXi

1

19n

::; n 2 exp{C n 2/

(lgn)1-2/p ~EXi ^}.

On the other hand, Lemma 1 and Kronecker lemma entail that 1 ~ EX~2

⁰

n 2/ p(lgn)l-2/p

Hence, the power of exp in the last expression of (7) is bounded by a 19 n( a > 0) for all sufficiently large n. Thus, choosing a ^{< a < 1, we}

have

00 n 00 1

L ^{P(L aniX: > E)} ~ CL ⁿ

< 00,

n=l i=l n=l

which implies (5) by Borel-Cantelli lemma.

Now we show that (6) holds. For the case

= ^1, note that

The first term on the last expression converges to CEIXII(IXI > N) a.s. by Kolmogorov SLLN. The second term clearly converges to O. The third term converges to 0 since EIXII(IXI > i/lgi) -- 0 as i --

Thus

n

limsup I L

^{X/, I} ~ ^{lim CEIXII(IXI} > N) = 0 a.s.,

n-+oo _ N-+oo

z=l

and so (6) holds when

= ^1. Next we assume that 1 < P < 2. By observing that

n . 1 . n

. max I

·X-"I <C · m a x · .

IX·"I

2k:=;n<2k+1

t:t

n 1/ p(lgn)1-1/p

~ ^{C (2k+1} )l/P(l~

^)l-l/p t; ^IX/'/,

we will obtain (6) if we show that

(8)

2^k

(2k)1/P(l~2k)1-1/P t; IX~'I-- ^{0 a.s.}

By Lemma 2, we have for any

> 0

00 2^k

(;P((2k)1/P(1~2k)1-1/P ~ ^{IX/'I > €)}

where

means that the summation is taken over all k such that 2k 2::

Henc; (8) follows by Borel-Cantelli lemma.

By replacing Xi by -Xi in the above argument we obtain that

n

1iminf~ ani(Xi - EXi)

2:: 0 a.s.

n--t(X) L...J

i=l

Thus the conclusion follows. o

The following theorem is an extension of Theorem

of Adler and Rosalsky [1]. It has less stringent condition than Theorem 1 when ani

aiJbn for 1

n and 1 <

< 2.

THEOREM

2. Let {X,Xn,n 2:: 1} be a sequence of LLd. random variables with EIXIP <

for some 1

p < 2. Let {an} and {bn}

be constants satisfying 0 < b

i

Assume that condition (2) holds.

Then

(i)

ai(Xi - EXi)/bn

0 a.s. if1 <

< 2.

(ii)

ai(Xi - EXi)/bn

0 a.s. if p

1 and condition (3)

holds.

Proof We need only to prove (i), since (ii) follows from Theorem 2 of Adler and Rosalsky [1]. Assume that 1 <

< 2. Define Y

=

XnI(IXn / :s: ⁿ

P) for n

1. From the proof of Theorem 2 of Adler and Rosalsky [1],

L~=l

ai(Xi - EYi) 0 b

n

--+

a.s.

The proof will be completed by showing that

aiE(Xi - Yi) 0

--+ •

b

By Kronecker lemma, it is enough to show that (9)

It

follows from condition (2) that

f,anE(Xn - Y n) I :s: f lanl EIXII(IXI > n

bn Ibnl

n=l n=l

= f I~nl f ^EIXII(i

^{< IXI :s: (i + l)l/p)}

n=l

I

i=n

00 i

I

= L E/XII(i

< IXI :s: (i + l)l/P) L I:n

i=l n=l

I nl

00 i

:s: _C~ EIXII(i

< IXI :s: ^{(i +l)l/p)} _~

. L...J . . . . L...J

i=l n=l

00

:s: CL i(P-l)/PEIXII(i

< IXI :s: (i + l)l/p)

i=l

00

:s: CL EIXIPI(i

P < IXI :s: ⁽ⁱ + ^l)l/p)

i=l

:s: CE/XIP <

which implies (9) and the proof is complete. o

COROLLARY

1. (Marcinkiewicz SLLN). Let {X,Xn,n

I} be a sequence of Li.d. random variables witb EIXIP <

for some 1

p <

2. Tben

Proof. Let an = 1 and bn =

for

1. Then condition (2) holds true. So Corollary 1 follows from Theorem 2. 0

The following corollary has been proved by Teicher [9].

COROLLARY

2. Let {X,Xn,n

I} be a sequence ofU.d. random variables witb EIXIP <

for some 1

p < 2. If {an} and {vn } are positive constants satisfying 0 <

i,

af

and

(10)

tben

Proof. Let bn

af)l/

for n

1. Under condition (10), (2) holds since

an an 1

b

-

(~n

^P)l/ =O(~/ ^).

i P n P

Also, condition (3) holds when

1 since

Thus, the conclusion follows from Theorem 2. o

The next example shows that Theorem 2(ii) can fail if condition (3)

is not assumed.

(11)

EXAMPLE.

Let {X, X

n 2:: 1} be a sequence of LLd. random vari- ables with probability density function

f(x)

_x

_ogx c

_'

(x),

< x <

where c is taken such that

(OO _ _ c _ _ dx =

12 x 2(1ogx)2 Then E[X]

cjlog2.

Let

l/n,O < ^b

i

and b

o(1og(1ogn)) for all n

l(for example,

log(1og(1og n) )). Let

be as in the proof of Theorem 2, Le., Y

X

1(IX

n) for n 2:: 1. Since 0 < ^b

i

we have

an

=

= O(~),

b

nb

n

hence (2) holds with

1 and it follows from the proof of Theorem 2 of Adler and Rosalsky [1] that

L~-l ai(Xi - EYi)

0 b

n

- ?

a.s.

But,

n n

1

:E ^{lail =} :E -;- '" ^logn,

i=l i=l 1.

which shows that condition (3) fails. Note that

-

_bL....Jt

a·E(X· - Y;).

t t

n i=l

1

1

11

= b :E ^{ai EX1(IXI > i) '" b} :E ^{i . xf(x) dx}

n i=l n i=l t

1

11

""-:E- _b

_i=2

_{i x(1og x)2} ^dx

=

~ t _1_ "" ~ in ¹ dx "" clog(1ogn).

b

i=2

log

b

i=2

log

b

Thus, noting that b

o(log(log n)), it follows that

(12)

1

-

"" a·E(X· -

1':)

Combining (11) ancl(12), wehave

References

1. A. Adler and A. Rosalsky, On the strong law of large numbers for normed weighted sums of i.i.d. random variables,Stochastic Anal. Appl. 5 (1987),467- 483.

2. B. D. Choi and S. H. Sung, Almost sure convergence theorems of weighted sums of random variables, Stochastic Anal. Appl. 5 (1987), 365-377.

3. , On moment conditions for the supremum of normed sums, Stochastic Process. Appl. 26 (1987), 99-106.

4. Y. S. Chow, Some convergence theorems for independent random variables, Ann. Math. Statistics 37 (1966), 1482-1493.

5. Y. S. Chow and T. L. Lai, Limiting behavior of weighted sums of independent random variables, Ann. Probability 1 (1973), 810-824.

6. L. T. Fernholz and H. Teicher, Stability of random variables and iterated log- arithm laws for martingales and quadratic forms, Ann. Probability 8 (1980), 765-774.

7. B. Jamison, S. Orey, and W. Pruitt, Convergence of weighted averages of in- dependent random variables, Z. Wahrsch. Verw. Gebiete4 (1965), 40-44.

8. W. F. Stout, Some results on the complete and almost sure convergence of lin- ear combinations of independent random variables and martingale differences, Ann. Math. Statistics 39(1968), 1549-1562.

9. H. Teicher, Almost certain convergence in double arrays, Z. Wahrsch. Verw.

Gebiete 69 (1985), 331-345.

10. R. Thrum, A remark on almost sure convergence of weighted sums, Probab.

Th. Rel. Fields 75 (1987), 425-430.

Department of Applied Mathematics Pai Chai University

Taejon 302-735, Korea