The strong laws of large numbers for weighted sums of pairwise quadrant dependent random variables

THE STRONG LAWS OF LARGE NUMBERS FOR WEIGHTED SUMS OF PAIRWISE QUADRANT

DEPENDENT RANDOM VARIABLES

TAE-SUNG KIM AND JONG IL BAEK

ABSTRACT. We derive the almost sure convergence for weighted sums of random variables which are either pairwise positive quadrant dependent or pairwise negative quadrant dependent and then apply this result to obtain the almost sure convergence of weighted aver- ages. We also extend some results on the strong law of large numbers for pairwise independent identically distributed random variables es- tablished in Petrov to the weighted sums of pairwise negative quad- rant dependent random variables.

1. Introduction

Many recent papers have been concerned with concepts of positive dependence and negative dependence for families of random variables (see for example Karlin and Rinott (1980), Block and Ting (1981), Ebrahimi and Ghosh (1981), Block, Savits and Shaked (1982) and the references therein). Lehmann (1966) introduced the notions of positive quadrant dependence and negative quadrant dependence: A sequence {Xi : i ~ I} of random variablesis called pairwise positive quadrant dependent(pairwise PQD) if for any realri, rj and i =Ij

(1) P{Xi >Ti, X j >Tj} ~ P{Xi>ri} P{Xj >Tj}

and it is called pairwise negative quadrant dependent (pairwise NQD) if for any real Ti, Tj andi =Ij

(2) P{Xi>Ti,Xj >rj}:::; P{Xi>Ti} P{Xj >rj}.

Received July25, 1997.

1991 Mathematics Subject Classification: 60F15, 60F99.

Key words and phrases: pairwise positive quadrant dependent, pairwise negative quadrant dependent, almost sure convergence, weighted sums.

This paper was supported by WonKwang University Grant in1998.

38 Tae-Sung Kim and JongnBaek

Let{Xi i _~ I} be a sequence of random variables, assumed throughout this article to be nondegenerate, and let {Wi : i _~ I} be a sequence of positive numbers. Define Sn =L:1 Wi Xi and Wn=L:1 Wi·

Etemadi(1983, b) already has studied the almost sure convergence of (Sn -ESn) / ^Wnto zero as n ^-+00, under restrictions (a) sUPi>1 EXi<

00 and (b) L:;:1 L:1 [Wi Wj Cov+(Xi, Xj)] / WJ ^< 00, for the case where {Xi : ⁱ ~ I} is a sequence of nonnegative random variables with finite second moments and {Wi : i ~ I} is a sequence of positive numbers satisfying

(3) Wn / ^Wn ^{-+ 0 and W}n ^{-+ 00as n -+ 00.}

In this paper we prove the almost sure convergence of(Sn - ESn) / ^Wn to zero as n -+ 00, for the case where the random variables are 'ei- ther pairwise positive quadrant dependent (PQD) or pairwise negative quadrant dependent (NQD). We also obtain analogues of "Kolmogorov's theorem" for weighted averages of pairwise positive (negative) quadrant random variables. These will ensure that the work of Etemadi (1983, ,b) on the almost sure convergence of weighted averages of pairwise in- dependent random variables remains validifthe assumption of pairwise independence is replaced by either pairwise positive quadrant depen- dence or pairwise negative quadrant dependence.

In Section 2 we study the strong law of large number for weighted sums of pairwise positively correlated nonnegative random variables and apply this result to the pairwise PQD random variables and in Section 3 we obtain a similar result for the weighted sums of pairwise nega- tively correlated nonnegative random variables and apply it to the pair- wise NQD random variables. Finally we extend some results on the strong laws of large numbers for the pairwise independent identically dis- tributed random variables established in Petrov (1996) to the weighted sums of pairwise NQD random variables with the common distribution.

2. PQD random variables

The following lemma is a modified version of Theorem1 of Etemadi (1983,b) to a sequence of pairwise nonnegatively correlated nonnegative random variables:

LEMMA 2.1. Let {Wi : i ~ 1} be a sequence of positive numbers satisfying (3) and let {Xi : i ~ 1} be a sequence of pairwise non- negatively correlated nonnegative random variables with finite second moments. Let Sn = 2:~1 Wi Xi. Assume

(a) sup EXi <00, i2:1

00

(b) L Wi COV(Xi, Si) / wl < 00.

i=l

Then as n-+ 00, (Sn - ESn) / Wn^-+ 0 a. s.

Proof First note that

00 j

L L WiWjCOV(Xi, X j)+/WJ j=l i=l

00 j

L L[WiWjCOV(Xi, Xj)]/wJ j=l i=l

00

L Wj(Xj , Sj)/wJ j=l

< ⁰⁰

since Cov(Xi, X j)+ = Cov(Xi, Xj) ~ 0for all i =1=j. Thus by Theorem

1 of Etemadi (1983,b) Lemma 2.1 isproved. 0

THEOREM 2.1. Let {Wi : i _~ 1} be a sequence of positive numbers satisfying (3) and let {Xi : i ~ 1} be a sequence of pairwise PQD random variables with finite second moments. Let Sn = 2::=1WiXi.

Assume

(a) sup EIXi - EXil < ^00,

i2:1

00

(b) L WiCOV(Xi, Si)/wl <^00.

i=l

Then as n -+ 00, (Sn - ESn)/Wn^-+ 0 a. s.

Proof. An equivalent condition to (1) is that

(4) Cov{f(Xi), g(Xj)) ~ 0

for all nondecreasing (nonincreasing) functions f and 9 such that the covariance exists (see Lemma 1 of Lehmann (1966)). Hence the ran- dom variables Xi - EXi , i ~ 1, are pairwise PQD and we may as- sume that for i _~ 1, EXi ⁼ O. It clear that {WiXi : i _~ I} is a

40 Tae-Sung Kim and JongnBaek

sequence of pairwise PQD random variables. We consider the sequence {WiX: : i;:::: 1} and its corresponding sum S~ = E~=lWiX:, For real t ^put I(t) = max{O,t}, get) = t. Then I, ^{9 -} I ^{and 9} +I ^are

nondecreasing. According to (4) there hold

0::;COV(f(WiXi) , l(wjXj

»

and

o < 12Cov((g - f)(WiXi), (g+l)(wjXj

»

+ 12Cov((g+f)(WiXi), (g - f)(wjXj

»

- COV(g(WiXi), g(wjXj

»-

(5) 0 ::; COV(WiX:, wjXf) ::; COV(WiXi, WjXj ).

It follows from (a), (b) here and (5) that {WiX: : i;:::: 1} is a sequence of pairwise nonnegatively correlated nonnegative random variables and that satisfies the conditions (a) and (b) of Lemma 2.1. Thus as n ^{- t}

'00, (S~- ES~)/Wn ^{- t}0a.s. A similar consideration for negative parts, say S~* = E~lWiX;, and the fact that ES~- ES~* = 0complete the

proof of Theorem 2.1. 0

REMARK. The strong law of large number for pairwise PQD ran- dom variables established in Birkel (1989) is the case where thew/s are identically one (see Theorem 1 of Birkel (1989».

From Theorem 2.1 Corollary 2.1 can be obtained:

COROLLAY 2.1. (Etemadi, 1983,b) Let {Wi :i ;:::: 1} be a sequence ofpositive number satisfying (3) and let {Xi : i ;:::: 1} be a sequence ofpairwise independent random variables with finite second moments.

Assume

(a) sup EIXi - EXil < 00,

i~l 00

(b)

L

i=l

LetSn =E~=l WiXi. Thenas n ^{- t}00, (Sn - ESn) / Wn^{- t}0 a. s.

Next, we introduce two applications of Theorem 2.1. The following corollaries are the almost sure convergence of weighted averages and

< 00.

weighted. logarithmic averages for pairwise PQD random variables re- spectively.

COROLLAY 2.2. Let {Xi : i ~ I} bea sequence of pairwise PQD random variables with Brute second moments and let Sn = E~l Xi.

Assume

(a) EX_i > 0 for alli,

(b) {EX_i : i ~ I} satisBes (3), (c) supEIXi - EXil < 00,

i2:1

(d)

f

Proof. Let Yn=Xn / EXn and ^Wn= EXn. Then {Yi : ⁱ ~ I} is a sequence of pairwise PQD random variables with EYi =1 and EY? < ⁰⁰ and {Wi : i ~ I} is a sequence of positive numbers satisfying (3) and Wn= ESn. Let Tn= 2::~1WiYi. Then Tn ⁼ Sn =2:::=1Xi. Thus from (d) we obtain

L

i=l

which yields, as n ---+ 00(Tn- ETn) / Wn ⁼ (Sn - ESn) / ESn - 0a,. s.

according to Theorem 2.1. This completes the proof. 0

COROLLAY 2.3. Let {Xi : i ~ I} be a sequence of pairwise PQD random variables with Enite second moments and let Sn = E:=l Xi·

Assume

(a) EXi > 0for alli,

(b) {EXi : i ~ I} satisBes (3), (c) sup EIXi - EXil < 00,

i;?:l

00 j

() " " Cov(Xi,Xj ) < ^00.

d L..JL..J ESiESj(logESj)2 j=l i=l

42 Tae-SungKimand Jong Il Baek

Thenas n.- 00, (logESnt¹ L:~=l(Xi/ESi ) - 1 a. s.

Proof Let Yn=_{X n / EXn}andWn⁼ EXn / ESn. ^Then {Yi : i 2': I}

is a sequence of pairwise PQD random variables and {Wi : i 2': I} is a sequence of positive numbers satisfying (3). From (b) it follows that Wn '" ^logESn(see the proof of Corollary 3 of Etemadi (1983, b)). Let Tn= L:~=lWiYi. Then it follows from (d) that

00 j 00

LL

2:

j=l i=l j=l

Now use Theorem 2.1 to get the desired result.

3. NQD random variables

The following lemma is an application of the strong law of large number for nonnegative random variables in Etemadi (1983, a) to the 'weighted sums of pairwise nonpositively correlated and nonnegative ran-

dom variables.

LEMMA 3.1. Let{Wi : i 2': I} be a sequence of positive numbers sat- isfying (3) and let {Xi : i ~ I} be a sequence of pairwise nonpositively correlated nonnegative random variables with finite second moments.

Assume

(a) supEXi < 00, i2:1

00

(b)

L

i=l

Let Sn = L:~lWiXi' Then as n - 00, (Sn - ESn) / ^Wn^{- 70}a. s.

Proof Let a > 1 and for each k 2': 1 set ^nk = inf{n : Wn 2': a^k}.

SinceWn / Wn+1 - 1 as n - ⁰⁰ it follows that Wn ^rv a^k for all large k. Therefore for some c > 0 and every i = 1,2,3, ... , {k : nk 2': i} c

{k : W_nk ~ W_{i }} c {k : ca^k 2': lVi} (see the proof of Theorem 1 in Etemadi (1983, b)). By Chebyshev's' inequality and by using proof of Theorem 1 in Etemadi (1983, a)

00 00

I:P{ISnk - ESnkl / Wnk >E} < b I:VarSnk / W~k

k=1 k=1

(6)

00 nk

< b

I:[I:

k=1 i=1

00

< bI:w;VarXi / wl

i=1

for every ^E >

°

°

(7) (Snk - ESnk ) / Wnk ~

°

Now given^n, a positive integer, for^nk ~ⁿ < nk+1

(8)

I

I

I

I

Wn - Wnk_1 W nk ^Wnk

by the monotonicity ofSn it follows from (a), (7) and (8) that (9) limsup(ISn - ESnl / Wn) Ssup(EXi)(a - 1)

i~l

for everya > 1 which concludes the proof. D LEMMA 3.2. (Birkel, 1992)Let X be a random variable with the finite second moment. Let X+ = max(X,O) and X- = max(-X, 0). Then

(a) Var(X+) ~ Var(X), (b) Var(X-) ~ Var(X).

THEOREM 3.1. Let {Wi : i ~ I} be a sequence of positive numbers satisfying (3) and let {w; : i ~ I} be a sequence of pairwise NQD random variables with finite second moments. Assume

(a) supEIX; - EXil < ^00,

i~1 00

(b)

L

i=l

44 Tae-Sung Kim and JongnBaek

Proof. First note that {WiXi : i ~ I} is a sequence of pairwise NQD random variable sinceWi ~ o. An equivalent condition to (2) is that (10)

for all nondecreasing (nonincreasing) functions f and 9 such that the covariance exists (see Lemma 1 of Lehmann (1966». Hence the ran- dom variables Xi - EXi, ^{i ~} 1, are pairwise NQD and we may assume that, for i ~ 1, EXi = O. We consider the sequence {WiXt : i ~

I} and its corresponding sum S~. For real t, put f(t) = max{O,t}.

Clearly, f(t) is a nondecreasing function. Thus according to (10) there holds COV(WiXt, wjX!) ~ 0, Le., {WiXt : i ~ I} is a sequence of nonpositively correlated nonnegative random variables. Since for i = 1,2,3,···, Var(wiXt) ~ Var(wiXi), according to Lemma 3.2, our assumptions (a) and (b) together with Lemma 3.1 imply that as n ---+ 00, (S~- ES~) / Wn ---+0a. s. A similar consideration for negative parts, sayS~* =L:1X i-, and the fact that ^ES~- ES~* =0 complete

the proof. 0

Now we consider the almost sure convergence of weighted averages for pairwise NQD random variables as an application of Theorem 3.1.

COROLLAY 3.1. Let {Xi : i ~ I} be a sequence of pairwise NQD random variables with finite second moments and let Sn = L~=lXi·

Assume

(a) EXi > 0for all i ~ 1, (b) {EXi : i _~ I} satisfies (3), (c) supE/Xi - EXil <00,

i~l

~ VarXi (d)

f::t

Proof Let Yn = Xn / EX_n and W_n = EXn , and use the proof of Corollary 2.2 and Theorem 3.1. Then the proof of Corollary 3.1 is com-

plete. 0

COROLLAY 3.2. Let {Xi : i ~ I} be a sequence of pairwise NQD random variables with finite second moments and let Sn = :E~1 Xi.

Assume

(a) EX_i > 0 for alli ~ 1, (b) {EX_i : i ~ I} satisfies(3), (c) supEIXi - EXi\< 00,

i~l

~ VarXi

(d)

f::t

Then as n ^--+ 00,

(11) _log1

(n

ESn ~ _ESi ^{--+ 1a. s.}

Proof. Let Yn=_{X n / EXn}andWn=_{EXn / ESn.} Clearly {Yi : i _~ I} is a sequence of pairwise NQD random variables and Wn '" 10gESn as in the proof of Corollary 2.3. Hence from (d) we have

(12)

00 ( x. ) ~ ^Var(wiYi)

z=Var E~. ^/(logESi)2 = L- W2

i=l t i=l t

00

=

z=

i=l

Let Tn = :E~=1WiYi. Then from (12) as n ^--+ 00 (Tn - ETn)/Wn ^--+

o^a.s. according to Theorem 3.1. SinceTn =:E~1WiYi =L~=l(XiiESi)

and ETn=L~=lWi =W n(11) follows. 0

Finally we extend some results on the strong law of large numbers of pairwise independent identically distributed random variables estab- lished by Petrov (1996) to the weighted sums of pairwise NQD random variables with the same distribution and derive necessary conditions of almost sure convergence ofSn/Wn to zero. First we recall the following version' of the Borel-Cantelli lemma (cf. Petrov (1975)).

46 Tae-Sung Kim and JongnBaek

LEMMA 3.3. (Matula, 1992) Let {An} be a sequence of events. If L:~=lP(An ) < 00, then P(An i.o) =0, _ifL:~=l P(An) =00,and p(An A_{j )} ~ P(Aj)P(A_{j )} for i =1= j, then P(A_n i.o.) = l.

LEMMA 3.4. (Lehmann, 1966) Let{Xi: i 2:: 1} be asequenceofpair- wise NQD random variablesandlet{9n} be a sequenceofnondecreasing (nonincreasing) functions 9n :R _~ R. Then {9n(Xn} ^isalso a sequence ofpairwise NQDrandom variables.

Let{Xi :i 2:: 1}be a sequence of random variables and let{Wi :i 2:: 1}

be a sequence of positive numbers called weight. DefineSn = L:~=lWiXi and Wn =L:~=lWi· We assume

(13) Wn/wnt 00 and W n^{~ 00} as n ~ 00.

In the following lemma we derive a necessary condition for almost sure convergence of weighted sums of pairwise NQD random variables with same distribution.

LEMMA 3.5. Let{Xi: i 2:: 1} beasequenceofpairwise NQD random variablesandlet{Wi :i 2:: 1} be asequenceofpositive numbers satisfying (13). Let Sn =L:~=lWiXi' If

(14) then (15)

L

n=l

Proof. From (14) and the equality

wnXn - - - - . - -Sn Sn-l W n-¹ W n W n Wn-¹ Wn it follows that

( ) 1VnXn

16 W

n

~ 0a. s.

sinceWn-tlWn~ 1. Put X: = max(O, X n) and X;; = max(O, -Xn).

Then it follows from (16) that 1VnX: jWn^~ 0 a. s. and wnX;; /Wn^~

oa. s. Since X: is a nondecreasing function ofXn , {X:}is a sequence of

pairwise NQD random variables by Lemma 3.4. Similarly, from the fact that {-Xn } is a sequence of pairwise NQD random variables {X;} is also a sequence of pairwise NQD random variables according to Lemma 3.4. Now define the events, for alln 2:: 1, An = {wnX; > lWn}, B n⁼ {wnX; > _~Wn}' Then for i i-j, we have

P(A;nAj ) ::;P(Ai)P(Aj ),

P(BinBj ) ::; P(B;)P(Bj ).

We apply Lemma 3.3: if2::1P(WnX; 2:: iW n)⁼ 00thenP( Ani.o.)

= 1 contrary to almost sure convergence ofwnX; / W n to O. Thus we get 2::=1P(wnX;t" 2:: iW n) < 00. The similar consideration for {X;}

yields 2::=1P(wnX; 2:: iWn) <00. Thus

L

n=l

L

n=l

00 1 00 1

<

L

L

n=l n=l

< 00,

and the proof is complete. o

Let f(x) be an even continuous function that is positive and strictly increasing in the region x > 0 and satisfying the condition f(x) ^-+ 00

as x -+ 00. We put

(17)

THEOREM 3.2. Let {Xi : i 2:: I} be a sequence of pairwise NQD random variables with same distribution. If(14) holds then

(18) E(f(Xr)) < 00.

Proof. Assumptions (14) and (17) with Lemma 3.5 imply the relation

48

00

Too-Sung Kim and JongnBaek

00

(19)

LP(lwnXnl ~ ^Wn) =

n=l

LP(IXll ~^Wn/wn)

n=l

00

L P(/Xll ~ i-l(n)) < 00.

n=l From (19) we obtain

00

(20)

L

n=l

For an arbitrary random variable Y the conditions 2:::=1P(IYI ~ n) < 00 and EIYI < 00 are equivalent. Therefore it follows from (~O)

that (18) holds. Thus the proof is complete. 0 ACKNOWLEDGEMENTS. The author wishes to thank the referee for the helpful comments to improve the draft.

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Department of Statistics Wonkwang University Chonbuk 570-749, Korea

E-mail: [email protected]