Complete Convergence of Weighted Sums for Arrays of Rowwise m-negatively Associated Random Variables

(School of Mathematics and Computer Science,Anhui Normal University, Wuhu,Anhui,241003)

Complete Convergence of Weighted Sums for Arrays of Rowwise m-negatively Associated Random Variables

GUO MING-LE,XU CHUN-YU AND ZHU DONG-JIN

(School of Mathematics and Computer Science,Anhui Normal University, Wuhu,Anhui,241003)

Communicated by Wang De-hui

In this paper,we discuss the complete convergence of weighted sums for arrays of rowwise m-negatively associated random variables.By applying moment inequality and truncation methods,the sufficient conditions of complete convergence of weighted sums for arrays of rowwise m-negatively associated random variables are established.These results generalize and complement some known conclusions.

complete convergence,negatively associated,m-negatively associated, weighted sum

1 Introduction

Let{Xn,n≥1}be a sequence of random variables.Hsu and Robbins[1]introduced the concept of complete convergence of{Xn}.A sequence{Xn,n=1,2,···}of random variables is said to converge completely to a constant C if

In view of the Borel-Cantelli lemma,this implies that Xn→ C almost surely.The converse is true if{Xn,n≥1}is a sequence of independent random variables.

De fi nition 1.1A fi nite family of random variables{Xi,1≤i≤n}is said to be negatively associated(NA,for short)if for every pair of disjoint subsetsAandBof{1,2,···,n}and any real nondecreasing coordinate-wise functionsf1onRAandf2onRB

wheneverf1andf2are such that covariance exists.

An in fi nite family of random variables{Xi,−∞＜i＜∞}is NA if every fi nite subfamily is NA.

The de fi nition of NA was introduced by Alam and Saxena[2]and was studied by Joag-Devet al.(see[3–4]).As pointed out and proved by Joag-Dev and Proschan[3],a number of wellknown multivariate distributions possess the NA property.Negative association has found important and wide applications in multivariate statistical analysis and reliability.Many investigators have discussed applications of negative association to probability,stochastic processes and statistics.

De fi nition 1.2Letm≥1be a fi xed integer.A sequence of random variables{Xi,i≥1}is said to bem-negatively associated(m-NA,for short)if for anyn≥2andi1,i2,···,insuch that|ik−ij|≥mfor all1≤k=j≤n,{Xi1,Xi2,···,Xin}is NA.

The m-NA random variables is a natural extension from NA random variables.Actually, the NA sequence is just the 1-NA sequence.Moreover,Huet al.[5]showed that there exists a sequence which is not NA but 2-NA.

Huet al.[6]proved a very general result for complete convergence of rowwise independent arrays of random variables which is stated in Theorem 1.1.

Theorem 1.1[6]Let{Xni,1≤i≤kn,n≥1}be an array of rowwise independent arrays of random variables.Suppose that for everyϵ＞0and someδ＞0,

Then

Huet al.[7]obtained the complete convergence of maximum partial sums for arrays of rowwise NA random variables by using an exponential inequality obtained by Shao[8]and their result is given in Theorem 1.2.

Theorem 1.2[7]Let{Xni,1≤ i≤ kn,n≥1}be an array of rowwise NA random variables such that the conditions(i)and(ii)in Theorem1.1are satis fi ed.Then

Kuczmaszewska[9]investigated complete convergence of weighted sums for arrays of rowwise NA random variables,and proved the following result.

Theorem 1.3[9]Let{Xni,i≥1,n≥1}be an array of rowwise NA random variables,{ani,i≥1,n≥1}be an array of real numbers,{bn,n≥1}be an increasing sequence of positive integers,and{cn,n≥1}be a sequence of positive real numbers.If for someq＞2,0＜t＜2and anyϵ＞0the following conditions are satis fi ed:

then

In this paper,we investigate the complete convergence for arrays of rowwise m-NA random variables which includes many previous results as corollaries.For example,Sunget al.[10]and Huet al.[6]investigated independent arrays of random variables and Huet al.[7]investigated rowwise NA arrays of random variables.We point out that in Theorem 2.1 of this paper we not only extends the result of Huet al.[7],but also provide di ff erent methods from those used by them.

2 Main Results and Some Lemmas

Now we state our main results.The proof will be given in Section 3.Throughout this paper, C represents a positive constant whose value may di ff erent at each appearance.The symbol I(A)denotes the indicator function of A,N denotes the positive integer set and[x]indicates the maximum integer not larger than x.Let{bn,n≥1}be an increasing sequence of positive integers,{cn,n≥1}be a sequence of positive real numbers,{Xni,1≤i≤bn,n≥1}be an array of rowwise m-NA random variables,and{ani,1≤i≤bn,n≥1}be an array of real numbers.

Theorem 2.1If for somet＞0,δ＞0and anyϵ＞0,the following conditions are satis fi ed:

(i)

(ii)there exists someq≥2such that

Remark 2.1Theorem 2.1 improves upon Theorem 1.3 of Kuczmaszewska[9].Moreover, from Theorem 2.1 we see that the condition(b)in Theorem 1.3 is unnecessary.

Corollary 2.1 extends the main result of Sunget al.[10]and can be obtained immediately from Theorem 2.1.

Corollary 2.1Under the conditions of Theorem2.1,in addition,if the following condition is satis fi ed:

then

(c)if the sequence{cn,n≥1}is not bounded away from zero,that is,ifand that

Then for allϵ＞0,

Remark 2.2It is obvious that if the sequence{cn,n≥1}is bounded away from zero, that is,ifthen the assumption(c)is unnecessary,which follows from the assumption(b).

Theorem 2.2If for somet＞0,δ＞0and anyϵ＞0,the following conditions are satis fi ed:

then

For the proof of the main results we need to restate a few lemmas for easy reference. The following lemmas play an important role in our main results.

Lemma 2.1[11]Let{Xi,1≤i≤n}be a fi nite family of NA mean zero random variables

with＜∞for every1≤i≤n,and setBn=Then for allϵ＞0,a＞0,

Lemma 2.2[8]Let{Xi,1≤i≤n}be a sequence of NA random variables with mean zero and＜∞for every1≤i≤n,1≤p≤2.Then

Lemma 2.3Let{Xi,i≥1}be a sequence ofm-NA random variables with mean zero and＜∞for everyi≥1,and

Then for alln≥m,x＞0,a＞0,

and

Proof.From(2.3)we can immediately get(2.4).Hence,to complete the proof,it is enough to show that(2.3)holds.

It is obvious from De fi nition 1.2}is a sequence of NA random variables for every 1≤j≤m,m≤n.Since

it follows from Lemma 2.1 that

So,(2.3)holds.

Lemma 2.4Let{Xi,1≤i≤n}be a sequence ofm-NA random variables with mean zero andE|Xi|p＜∞for every1≤i≤n,1≤p≤2.ThenE

Proof.Let Yi,Tmk+jand r be as in Lemma 2.3.By using the Crinequality,it follows from Lemma 2.2 that

3 Proofs of the Main Results

Proof of Theorem 2.1Let

where δ＞0 and 1≤i≤bn,n≥1.By Property 6 in[3],we can conclude that

is an array of rowwise m-NA random variables.For n≥1 and 1≤k≤bn,let

Noting that for any n≥1,

Therefore,we have

Using Markov’s inequality,we get

Combining condition(i)with(3.1)–(3.3)we see that,to complete the proof,it is enough to show that

Set

For any ϵ＞0 and a＞0,set

where N={1,2,3,···}.Note

Hence it suffices to prove that

By Lemma 2.3,we have

Note that for any n∈A,

Thus,for any n∈A,we have

Therefore,by(3.4)and(3.5),the proof will be completed if we show that

we have

Therefore(2.1)holds.

Proof of Corollary 2.2Note that

Since EXni=0,it follows that

Thus,by(3.6),(3.7),(a),(b)and(c),we see that the conditions of Corollary 2.1 are satis fi ed. So,by Corollary 2.1 we complete the proof of Corollary 2.2.

Proof of Theorem 2.2Let Yni,be as in the proof of Theorem 2.1.From the proof of Theorem 2.1,we need only to prove that

holds.

In fact,using the Crinequality,for any r＞0,we can estimate

Thus,using Markov’s inequality,by the above estimation and(2.5)we obtain

Therefore,from the conditions(i),(ii)and(3.9),we know that(3.8)holds.

[1]Hsu P L,Robbins H.Complete convergence and the law of large numbers.Proc.Nat.Acad. Sci.U.S.A.,1947,33(2):25–31.

[2]Alam K,Saxena K M L.Positive dependence in multivariate distributions.Comm.Statist. Theory Methods,1981,10(12):1183–1196.

[3]Joag-Dev K,Proschan F.Negative association of random variables with applications.Ann. Statist.,1983,11(1):286–295.

[4]Block H W,Savits T H,Sharked M.Some concepts of negative dependence.Ann.Probab., 1982,10(3):765–772.

[5]Hu Y J,Ming R X,Yang W Q.Large deviations and moderate deviations for m-negatively associated random variables.Acta Math.Sci.(English Ed.),2007,27(4):886–896.

[6]Hu T C,Szynal D,Volodin A.A note on complete convergence for arrays.Statist.Probab. Lett.,1998,38(1):27–31.

[7]Hu T C,Volodin A.A Remark on Complete Convergence for Arays of Rowwise Negatively Associated Variables.in:Proceedings of the 3rd Sino-International Symposium on Probability, Statistics,and Quantitative Management.Taipei,Taiwan,Republic of China:ICAQM/CDMS, 2006:9–18.

[8]Shao Q M.A comparison theorem on moment inequalities between negatively associated and independent random variables.J.Theoret.Probab.,2000,13(2):343–356.

[9]Kuczmaszewska A.On complete convergence for arrays of rowwise negatively associated random variables.Statist.Probab.Lett.,2009,79(1):116–124.

[10]Sung S H,Volodin A,Hu T C.More on complete convergence for arrays.Statist.Probab.Lett., 2005,71(4):303–311.

[11]Liu L X,Wu R.Inequalities of maximal partial sum and the laws of iterated logarithm for sequence of NA random variables.Acta Math.Sinica,2002,45(5):969–978.

tion:60F15

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1674-5647(2014)01-0041-10

Received date:April 20,2011.

Foundation item:The NSF(10901003)of China,the NSF(1208085MA11)of Anhui Province and the NSF (KJ2012ZD01)of Education Department of Anhui Province.

E-mail address:mlguo@mail.ahnu.edu.cn(Guo M L).