Aiting SHEN Mei YAO Benqiong XIAO

1 Introduction

It is well known that complete convergence plays a very important role in the probability limit theory and mathematical statistics,especially in establishing the strong convergence rate for partial sums of random variables.The concept of complete convergence was introduced by Hsu and Robbins[1]as follows.

Definition 1.1A sequence{Un,n ≥1}of random variables is said to converge completelyto a constantaif for anyε＞

In this case,we write Un→a completely.In view of the Borel-Cantelli lemma,this implies that Un→a almost surely(a.s.,in short).The converse is true if random variables{Un,n≥1}are independent.Hsu and Robbins[1]proved that the arithmetic means of independent and identically distributed(i.i.d.,in short)random variables converges completely to the expected value if the variance of the summands is finite.Erdö s[2]proved the converse.The result of Hsu-Robbins-Erdö s is a fundamental theorem in probability theory and has been extended in several directions by many authors.One of the most important generalizations was provided by Baum and Katz[3]for the strong law of large numbers as follows.

Theorem 1.1Let＜ α ≤ 1andαp＞ 1.Let{Xn,n≥ 1}be independent and identically distributed random variables with zero means.Then the following statements are equivalent:

Up to now,there have been many versions of the Baum-Katz-type results for independent and dependent random variables,such as Gut[4],Peligrad and Gut[5],Wang et al.[6],Shen and Wu[7],and so on.

Chow[8]generalized the concept of complete convergence and introduced the concept of complete moment convergence,which is more general than complete convergence.Let{Zn,n≥1}be a sequence of random variables,and∞ for all ε＞ 0,then the above result is called the complete moment convergence.

Chow[8]obtained the following result on the complete moment convergence for i.i.d.random variables.

Theorem 1.2Suppose that{Xn,n≥1}is a sequence ofi.i.d.random variables withEX1=0, α ＞,p≥ 1andαp＞ 1.IfE[|X1|p+|X1|log(1+|X1|)]＜ ∞,then for allε＞ 0,

Since Chow[8]established the result of Theorem 1.2 for i.i.d.random variables,many authors have studied this type of complete moment convergencefor dependent random variables.See,for example,Chen and Wang[9]for the ϕ-mixing sequence,Wu et al.[10]and Wang et al.[11]for eρ-mixing sequence and the martingale difference sequence,respectively,and so on.

We should point out that the key techniques used in the proofs of Theorems 1.1–1.2 are the Rosenthal-type maximal moment inequality and the truncation methods.All the literatures above adopted these approaches or added extra conditions.There are many sequences of random variables satisfying the Rosenthal-type maximal moment inequality,such as independent random variables,negatively associated random variables,negatively supperadditive dependent random variables,ϕ-mixing random variables, eρ-mixing random variables,asymptotically almost negatively associated random variables,and so on.But negatively orthant dependent random variables and extended negatively dependent random variables do not satisfy the Rosenthal-type maximal moment inequality.If we want to generalize the results of Theorems 1.1–1.2 for i.i.d.random variables to the case of the extended negatively dependent setting,we should use different methods.The main purpose of this paper is to generalize the results of Theorems 1.1–1.2 for i.i.d.random variables to the case of the extended negatively dependent setting without identical distribution.In addition,we will present the sufficient and necessary conditions of complete moment convergence for extended negatively dependent random variables.

Now,let us recall the definition of extended negatively dependent random variables.

Definition 1.2Afinite collection of random variablesX1,X2,···,Xnis said to be extended negatively dependent(END,in short),if there exists a constantM＞0such that both

and

hold for all real numbersx1,x2,···,xn.An infinite sequence{Xn,n ≥ 1}is said to be END if everyfinite subcollection is END.

An array of random variables{Xni,1≤i≤n,n≥1}is called rowwise END random variables if for everyn≥1,{Xni,1≤i≤n}are END random variables.

The concept of the END sequence was introduced by Liu[12].In the case M=1,the notion of END random variables reduces to the well-known notion of the so-called negatively orthant dependent(NOD,in short)random variables,which was introduced by Joag-Dev and Proschan[13].They also pointed out that the negatively associated(NA,in short)random variables are NOD and thus NA random variables are END.Hence,the class of END includes the independent sequence,the NA sequence and the NOD sequence as special cases.Studying the limiting behavior of END random variables is of great interest.

Some applications for the END sequence have been found.See,for example,Chen et al.[14]established the strong law of large numbers for extend negatively dependent random variables and showed its applications to risk theory and renewal theory;Shen[15]presented some probability inequalities for END sequences and gave some applications;Wu and Guan[16]presented some convergence properties for the partial sums of END random variables;Wang and Wang[17]investigated a more general precise large deviation result for random sums of END real-valued random variables in the presence of consistent variation;Qiu et al.[18]and Wang et al.[19–21]provided some results on complete convergence for sequences of END random variables or arrays of rowwise END random variables;Wang et al.[22]studied the complete consistency for the estimator of nonparametric regression models based on END errors,and so on forth.The main purpose of the paper is to generalize the results of Theorems 1.1–1.2 for i.i.d.random variables to the case of the END setting,and the sufficient and necessary conditions of complete moment convergence for END random variables will also be established.

This work is organized as follows:Some important lemmas are provided in Section 2.The main results and their proofs are presented in Section 3.

Throughout this paper,the symbol C denotes a positive constant which is not necessarily the same in each appearance,and an=O(bn)stands for an=C(bn).I(A)is the indicator function of an event A.Denote logx=lnmax(x,e).

2 Preliminaries

In this section,we will provide some important lemmas,which will be applied to prove the main results of this paper.Thefirst one is a basic property for END random variables,which was given by Liu[23].

Lemma 2.1Let random variablesX1,X2,···,Xnbe END.Iff1,f2,···,fnare all nondecreasing(or nonincreasing)functions,then random variablesf1(X1),f2(X2),···,fn(Xn)are END.

The next one is the Marcinkiewicz-Zygmund-type inequality and the Rosenthal-type inequality for partial sums and maximum partial sums of END random variables.

Lemma 2.2Letp≥1and{Xn,n≥1}be a sequence of END random variables withEXi=0andE|Xi|p＜∞for eachi≥1.Then there exists a positive constantCpdepending only onpsuch that

ProofThe inequality(2.3)has been established by Shen[15].The inequality(2.1)can be obtained in a similar way as that of Corollary 2.2 in Asadian et al.[24].The inequalities(2.2)and(2.4)can be proved by using the inequalities(2.1),(2.3)in a similar way as that of Theorem 2.3.1 in Stout[25],respectively.The details of the proof are omitted.

With Lemma 2.2 accounted for,we can get the following important property for END random variables,which will play an important role in proving the main results of this paper.The proof is similar to that of Lemma A6 in Zhang and Wen[26],so the details are omitted.

Lemma 2.3Let{Xn,n≥1}be a sequence of END random variables.Then there exists a positive constantCsuch that for anyx≥0and alln≥1,

The following is a basic property for stochastic domination.For the proof,one can refer to Wu[27],or Wang et al.[6].

Lemma 2.4Let{Xn,n≥1}be a sequence of random variables,which is stochastically dominated by a random variableX,i.e.,there exists a positive constantCsuch that

for allx≥0andn≥1.Then for anyα＞0andb＞0,the following two statements hold:

whereC1andC2are positive constants.Consequently,E|Xn|α≤ CE|X|α,whereCis a positive constant.

The last one comes from Sung[28].

Lemma 2.5LetYn,Zn,n≥1be random variables.Then for anyq＞1,ε＞0anda＞0,

3 Main Results and Their Proofs

In this section,we will give the main results of this paper,including the sufficient and necessary conditions of complete convergence and complete moment convergence for END random variables.

3.1 Sufficient and necessary conditions for complete convergence

Theorem 3.1Letα ＞andαp ＞ 1.Let{Xn,n ≥ 1}be a sequence of END random variables withEXn=0ifp≥1.If there exists a random variableXand two positive constantsC1andC2such that

for allx≥0andn≥1,then the following statements are equivalent:

(i)E|X|p＜∞;

(ii)for allε＞ 0,

Proof(ii)⇒ (i)is trivial.So it suffices to show(i)⇒ (ii).We consider the following two cases.

Case 10＜p＜1.

Forfixed n≥1,denote,for 1≤i≤n,that

Noting that Xi=Yni+Zni,we have that for all ε＞ 0,

It follows from Markov’s inequality,Crinequality and Lemma 2.4,that

Hence,the desired result(3.2)follows from(3.3)–(3.5)immediately.

Case 2p≥1.

Noting that αp ＞ 1,we take a suitable q such that＜q＜1.Forfixed n≥1,denote for 1≤i≤n that

Noting that

for 1≤ j≤ n,we have that for all ε＞ 0,

Hence,in order to prove(3.2),it suffices to show that I1＜∞,I2＜∞and I3＜∞.

For I1,wefirstly show that

It follows from EXn=0,Markov’s inequality and Lemma 2.4,that

which together with E|X|p＜∞and＜q＜1 yields(3.7).Hence,by(3.7),we have that

Forfixed n≥ 1,we can see thatare still END random variables by Lemma 2.1.It follows from(3.8),Markov’s inequality and Lemma 2.2,that for any δ≥ 2,

It follows from Crinequality,Markov’s inequality and Lemma 2.4,that

Hence,I1＜ ∞ follows from(3.9)–(3.11)immediately.

In the following,we will show that I2＜∞.Forfixed n≥1,denote,for 1≤i≤n,that

It is easily checked that

which implies that

It follows from(3.1)and E|X|p＜∞,that

Forfixed n≥ 1,we can see that1≤i≤n}are still END random variables by Lemma 2.1.It follows from Markov’s inequality,Crinequality and Lemma 2.2,that

By Crinequality and Lemma 2.4,we can get that

Similarly to the proof of(3.11)and(3.17),we can obtain that J2＜∞,which,together with(3.16)and(3.17),yields that I2＜∞.

Similarly to the proof of I2＜∞,one can get that I3＜∞.Hence,(3.2)follows from(3.6),I1＜∞,I2＜∞and I3＜∞immediately.This completes the proof of the theorem.

With Theorem 3.1 accounted for,we can get the Marcinkiewicz-Zygmund-type strong law of large numbers for END random variables without identical distribution as follows.

Corollary 3.1Letα ＞andαp ＞ 1.Let{Xn,n ≥ 1}be a sequence of END random variables withEXn=0ifp≥1.Assume that there exists a random variableXand twopositive constantsC1andC2such that(3.1)holds for allx≥0andn≥1.IfE|X|p＜∞,then

ProofSince E|X|p＜ ∞,by Theorem 3.1,we have that for all ε＞ 0,

It follows from(3.19)that,for all ε＞ 0,

which,together with the Borel-Cantelli lemma,yields that

For all positive integers n,there exists a positive integer k such that 2k−1≤n ≤2k.We have,by(3.20),that

which implies(3.18).This completes the proof of the corollary.

If{Xn,n≥1}is a sequence of END random variables with identical distribution,then(3.1)is obvious.By using Theorem 3.1,we can get the Baum-Katz-type result for END random variables as follows.

Corollary 3.2Letα ＞andαp ＞ 1.Let{Xn,n ≥ 1}be a sequence of END random variables with identical distribution.Assume further thatEX1=0ifp≥1.Then(i)and(ii)in Theorem3.1are equivalent.

3.2 Sufficient and necessary conditions for complete moment convergence

In this subsection,we will establish the sufficient and necessary conditions for complete moment convergence.First we present the necessary condition for complete moment convergence.

Theorem 3.2Suppose that the conditions of Theorem3.1hold.If for allε＞ 0,

thenE|X|p＜∞.

ProofNote that

Combining(3.21)and(3.22),we can get that(3.2)holds for all ε＞ 0.Hence,E|X|p＜ ∞follows from Theorem 3.1 immediately.This completes the proof of the theorem.

Next we present the sufficient condition for complete moment convergence.Noting that the factor logn is added to the right of the maximal moment inequality for END random variables(see Lemma 2.2),in order to establish the sufficient condition for complete moment convergence,the moment condition should be changed.

Theorem 3.3Suppose that the conditions of Theorem3.1hold forp≥ 1.IfE|X|plogθ|X|

ProofForfixed n≥ 1,denote,for 1≤ i≤ n,that

It follows from Lemma 2.5,that for any δ＞ 1,

Noting that|Zni|=(|Xi|−nα)I(|Xi|＞ nα)≤ |Xi|I(|Xi|＞ nα),we have,by Lemma 2.4,that

Next,we will show that P1＜ ∞.Noting that θ＞ p ≥ 1,we can take δ= θ.We consider the following two cases.

Case 11＜θ≤2.

It follows from(2.2)of Lemma 2.2 and Lemma 2.4,that

Case 2θ＞ 2.

Hence,the desired result(3.23)follows from(3.24)–(3.27)immediately.This completes the proof of the theorem.

AcknowledgementsThe authors are most grateful to the editor and anonymous referees for their careful reading of the manuscript and many valuable suggestions which helped to improve an earlier version of this paper.

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