Complete Convergence for Weighted Sums of WOD Random Variables

ZHANG Ying，ZHANG Yu，SHEN Ai-ting

（School of Mathematical Sciences，Anhui University，Hefei 230601，China）



Complete Convergence for Weighted Sums of WOD Random Variables

ZHANG Ying，ZHANG Yu，SHEN Ai-ting

（School of Mathematical Sciences，Anhui University，Hefei 230601，China）

In this article，we study the complete convergence for weighted sums of widely orthant dependent random variables.By using the exponential probability inequality，we establish a complete convergence result for weighted sums of widely orthant dependent random variables under mild conditions of weights and moments.The result obtained in the paper generalizes the corresponding ones for independent random variables and negatively dependent random variables.

widely orthant dependence；complete convergence；weighted sums

2000 MR Subject Classification：60F15

Article ID：1002—0462（2016）01—0001—08

Chin.Quart.J.of Math.

2016，31（1）：1—8

§1.　Introduction

It is well known that the probability limit theorem and its applications for independent random variables have been studied by many authors，while the assumption of independence is not reasonable in real practice.If the independent case is classical in the literature，the treatment of dependent random variables is more recent.

One of the important dependence structure is the wide dependence structure，which was introduced by Wang et al（2013）as follows.

Definition 1.1For the random variables｛Xn，n≥1｝，if there exists a finite real sequence｛gU（n），n≥1｝satisfying for each n≥1 and for all xi∈（-∞，∞），1≤i≤n，

then we say that the｛Xn，n≥1｝are widely upper orthant dependent（WUOD，in short）；if there exists a finite real sequence｛gL（n），n≥1｝satisfying for each n≥1 and for all xi∈（-∞，∞），1≤i≤n，

then we say that the｛Xn，n≥1｝are widely lower orthant dependent（WLOD，in short）；if they are both WUOD and WLOD，then we say that the｛Xn，n≥1｝are widely orthant dependent（WOD，in short）and gU（n），gL（n），n≥1，are called dominating coefficients.

An array｛Xni，i≥1，n≥1｝of random variables is called rowwise WOD if for every n≥1，｛Xni，i≥1｝is a sequence of WOD random variables.

Recall that when gL（n）=gU（n）=M for some constant M，the random variables｛Xn，n≥1｝are called extended negatively upper orthant dependent（ENUOD，in short）and extended negatively lower orthant dependent（ENLOD，in short），respectively.If they are both ENUOD and ENLOD，then we say that the random variables｛Xn，n≥1｝are extended negatively orthant dependent（ENOD，in short）.The concept of general extended negative dependence was proposed by Liu（2009）and further promoted by Chen et al（2010），Wang and Wang（2013），Wang et al（2014a）and so forth.When gL（n）=gU（n）=1 for any n≥1，the random variables｛Xn，n≥1｝are called negatively upper orthant dependent（NUOD，in short）and negatively lower orthant dependent（NLOD，in short），respectively.If they are both NUOD and NLOD，then we say that the random variables｛Xn，n≥1｝are negatively orthant dependent（NOD，in short）.

The concept of NOD was introduced by Joag-Dev and Proschan（1983）.For more details about NOD random variables，one can refer to Wang et al（2010，2011），Wu（2010），Wu and Jiang（2011），Sung（2011），Qiu et al（2011）and so forth.Joag-Dev and Proschan（1983）pointed out that NA random variables are NOD.By the statements above，we can see that the class of WOD random variables contains END random variables，NOD random variables，NA random variables and independent random variables as special cases.Hence，studying the probability limiting behavior of WOD random variables and its applications are of great interest.

The concept of WOD random variables was introduced by Wang et al（2013）and many applications have been found subsequently.See，for example，Wang et al（2013）provided some examples which showed that the class of WOD random variables contains some common negatively dependent random variables，some positively dependent random variables and some others.He et al（2013）provided the asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables.Chen et al（2013）considered uniform asymptotic for the finite-time ruin probabilities of two kinds of nonstandard bidimensional renewal risk models with constant interest forces and diffusion generated by Brownian motions.Shen（2013）established the Bernstein type inequality for WOD random variables and gave some applications.Wang et al（2014b）studied the complete convergence for weighted sums of arrays of rowwise WOD random variables and gave its applications to nonparametric regression models and so forth.The main purpose of the paper is to further study the complete convergence for weighted sums of arrays of rowwise WOD random variables.

The definition of stochastic domination will be used in the paper as follows.

Definition 1.2An array｛Xni，i≥1，n≥1｝of rowwise random variables is said to be stochastically dominated by a random variable X if there exists a positive constant C such that

for all x≥0，i≥1 and n≥1.

§2.　Main Result and Its Proof

In order to prove Theorem 2.1，we need the following two important lemmas.The first one is a basic property for WOD random variables which can be found in Wang et al（2014b）.

Lemma 2.1Let｛Xn，n≥1｝be a sequence of WOD random variables.

（i）If｛fn（·），n≥1｝are all nondecreasing（or all nonincreasing），then｛fn（Xn），n≥1｝are still WOD.

（ii）For each n≥1 and any s∈R，

The next one is a fundamental inequality for stochastic domination.For the proof，one can refer to Wu（2006），Wu and Wu（2011）or Wang et al（2012）.

Lemma 2.2Let｛Xni，i≥1，n≥1｝be an array of rowwise random variables which is stochastically dominated by a random variable X.For any α＞0 and b＞0，the following two statements hold

where C1and C2are positive constants.Consequently，E|Xni|α≤CE|X|α，where C is a positive constant.

With Lemma 2.1 and Lemma 2.2 accounted for，we can give the proof of Theorem 2.1.

and

So，without of loss generality，we may assume that ank＞0.Moreover，we may assume that K≥1 and An＞0 for each n.For 0＜β＜α＜2β and N=2，3，···，let

It follows that

Since｛Ynk，k≥1，n≥1｝are monotone transformations of｛Xnk，k≥1，n≥1｝，｛Ynk，k≥1，n≥1｝is an array of rowwise WOD random variables by Lemma 2.1（i）.Moreover，｛ankYnk，k≥1，n≥1｝is an array of rowwise WOD random variables，since ank＞0.

We first show that I1＜∞.If a random variable X≤1 a.e.，then EeX≤eEX+EX2.Let 0＜t＜K-1n-β，then tankYnk/An≤1，E（ankYnk）＜0.Hence，we have by Lemma 2.2 that

where

Putting t=n-β（2C）-1，by An≤Kn-α，we have for sufficiently large n that

Noting that g（n）=O（nδ）for some δ＞0 and 0＜β＜α，we have

On the other hand，by the definition of stochastic domination，we have

Since E|X|2/α＜∞，N，C and K are fixed constants，using Markov’s inequality，we can easily obtain

From（2.1）～（2.3），we have

Noting that，｛-Xnk，k≥1，n≥1｝is also WOD by Lemma 2.1（i），we obtain

Hence the proof is complete.

Remark 3.1Noting that NA，NOD，END and independent random variables are all WOD，hence，the main result of the paper still holds for these sequences.

Acknowledgements The authors are most grateful to the Editor and anonymous referees for careful reading of the manuscript and valuable suggestions which helped to improve an earlier version of this paper.

［References］

［1］CHEN Yi-qing，CHEN An-yue，NG K W.The strong law of large numbers for extended negatively dependent random variables［J］.Journal of Applied Probability，2010，47：908-922.

［2］CHEN Yang，WANG Le，WANG Yue-bao.Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models［J］.Journal of Mathematical Analysis and Applications，2013，401：114-129.

［3］HE Wei，CHENG Dong-ya，WANG Yue-bao.Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables［J］.Statistics and Probability Letters，2013，83：331-338.

［4］HSU P L，ROBBINS H.Complete convergence and the law of large numbers［J］.Proceedings of the National Academy of Sciences of the United States of America，1947，33（2）：25-31.

［5］JOAG-DEV K，PROSCHAN F.Negative association of random variables with applications［J］.Annals of Statistics，1983，11（1）：286-295.

［6］LIU Li.Precise large deviations for dependent random variables with heavy tails［J］.Statistics and Probability Letters，2009，79：1290-1298.

［7］QIU De-hua，CHANG K C，GIULIANO A R，et al.On the strong rates of convergence for arrays of rowwise negatively dependent random variables［J］.Stochastic Analysis and Applications，2011，29：375-385.

［8］SHEN Ai-ting.Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models［J］.Abstract and Applied Analysis，2013，2013（1）：1-9，Article ID 862602.

［9］SUNG S H.On the exponential inequalities for negatively dependent random variables［J］.Journal of Mathematical Analysis and Applications，2011，381：538-545.

［10］WANG Kai-yong，WANG Yue-bao，GAO Qing-wu.Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate［J］.Methodology and Computing in Applied Probability，2013，15：109-124.

［11］WANG Shi-jie，WANG Xue-jun.Precise large deviations for random sums of END real-valued random variables with consistent variation［J］.Journal of Mathematical Analysis and Applications，2013，402：660-667.

［12］WANG Xue-jun，HU Shu-he，YANG Wen-zhi，et al.Exponential inequalities and inverse moment for NOD sequence［J］.Statistics and Probability Letters，2010，80：452-461.

［13］WANG Xue-jun，HU Shu-he，YANG Wen-zhi.Some convergence results for arbitrary sequences under moment condition［J］.Chinese Quarterly Journal of Mathematics，2011，26（4）：585-589.

［14］WANG Xue-jun，HU Shu-he，YANG Wen-zhi.Complete convergence for arrays of rowwise negatively orthant dependent random variables［J］.RACSAM，2012，106（2）：235-245.

［15］WANG Xue-jun，LI Xiao-qin，HU Shu-he，et al.On complete convergence for an extended negatively dependent sequence［J］.Communications in Statistics-Theory and Methods，2014a，43：2923-2937.

［16］WANG Xue-jun，XU Chen，HU T-C，et al.On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models［J］.TEST，2014b，23：607-629.

［17］WU Qun-ying.Probability Limit Theory for Mixing Sequences［M］.Beijing：Science Press of China，2006.

［18］WU Qun-ying.A strong limit theorem for weighted sums of sequences of negatively dependent random variables［J］.Journal of Inequalities and Applications，2010，2010（1）：1-8，Article ID 383805.

［19］WU Qun-ying，JIANG Yuan-ying.The strong consistency of M estimator in a linear model for negatively dependent random samples［J］.Communications in Statistics-Theory and Methods，2011，40：467-491.

［20］WU Yan-chun，WU Qun-ying.Complete convergence properties of the sums for ρ-mixing sequences of random variables［J］.Chinese Quarterly Journal of Mathematics，2011，26（2）：159-163.

O211.4Document code：A

date：2014-07-08

Supported by the Natural Science Foundation of Anhui Province（1308085QA03）；Supported by the Quality Improvement Projects for Undergraduate Education of Anhui University（ZLTS2015035）；Supported by the Research Teaching Model Curriculum of Anhui University（xjyjkc1407）；Supported by the Students Innovative Training Project of Anhui University（201410357118）

Biographies：ZHANG Ying（1993-），female，native of Putian，Fujian，engages in probability limit theorem；ZHANG Yu（1989-），male，native of Hefei，Anhui，engages in probability limit theorem；SHEN Ai-ting（1979-），female，native of Hefei，Anhui，an associate professor of Anhui Universiry，Ph.D.，engages in probability limit theorem.