Complete Moment Convergence for Negatively Dependent Sequences of Random Variables QunyingWu and

We study the complete moment convergence for sequences of negatively dependent identically distributed random variables with , , , and , . As a result, we establish the new complete moment convergence theorems.


Introduction and Main Results
Random variables  and  are said to be negative quadrant dependent (NQD) if  ( ≤ ,  ≤ ) ≤  ( ≤ )  ( ≤ ) for all ,  ∈ R. A collection of random variables is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the collection satisfies (1).
It is important to note that (1) implies  ( > ,  > ) ≤  ( > )  ( > ) for all ,  ∈ R.Moreover, it follows that (2) implies (1) and, hence, (1) and ( 2) are equivalent.However, Ebrahimi and Ghosh [1] showed that (1) and ( 2) are not equivalent for a collection of 3 or more random variables.Accordingly, the following definition is needed to define sequences of negatively dependent random variables.( An infinite sequence of random variables {  ;  ≥ 1} is said to be ND if every finite subset  1 , . . .,   is ND. Definition 2. Random variables  1 ,  2 , . . .,   ,  ≥ 2, are said to be negatively associated (NA) if, for every pair of disjoint subsets  1 and  2 of {1, 2, . . ., }, where  1 and  2 are increasing for every variable (or decreasing for every variable) function such that this covariance exists.A sequence of random variables {  ;  ≥ 1} is said to be NA if its every finite subfamily is NA.
The definition of PNQD is given by Lehmann [2].The definition of NA is introduced by Joag-Dev and Proschan [3], and the concept of ND is given by Bozorgnia et al. [4].These concepts of dependent random variables are very useful for reliability theory and applications.
It is easy to see from the definitions that NA implies ND.But Example 1.5 in Wu and Jiang [5] shows that ND does not imply NA.Thus, it is shown that ND is much weaker than NA.In the articles listed earlier, a number of wellknown multivariate distributions are shown to possess the ND properties.In many statistics and mechanic models, a ND assumption among the random variables in the models is more reasonable than an independent or NA assumption.Because of wide applications in multivariate statistical analysis and reliability theory, the notions of ND random variables have attracted more and more attention recently.A series of useful results have been established (cf.Bozorgnia et al. [4], Fakoor and Azarnoosh [6], Asadian et al. [7], Wu [5,8], Wang et al. [9], and Liu et al. [10]).Hence, it is highly desirable and of considerable significance in the theory and application to study the limit properties of ND random variables theorems and applications.
Chow [11] first investigated the complete moment convergence, which is more exact than complete convergence.Thus, complete moment convergence is one of the most important problems in probability theory.The recent results can be found in Chen and Wang [12], Gut and Stadtmüller [13], Sung [14], Guo [15], and Qiu and Chen [16,17].In addition, Qiu and Chen [17] obtained complete moment convergence theorems for independent identically distributed sequences of random variables with E = 0, E exp(ln  ||) < ∞,  > 1.A natural question is whether there is any type of complete moment convergence theorems for 0 <  ≤ 1.
In this paper, we study the complete moment convergence for sequences of negatively dependent identically distributed random variables with E = 0, E exp(||  ) < ∞, 0 <  < 1, and E exp(||ln − ||) < ∞,  > 0. As a result, we establish the new complete moment convergence theorems.
In the following, the symbol  stands for a generic positive constant which may differ from one place to another.Let   ≪   denote that there exists a constant  > 0 such that   ≤   for sufficiently large , ln  mean ln(max(, )), and  denotes an indicator function.

Proofs
The following four lemmas play important roles in the proof of our theorems.
(i) Let {  ;  ≥ 1} be a sequence of Borel functions; all of them are monotone increasing (or all are monotone decreasing).Then, {  (  );  ≥ 1} is a sequence of ND r.v.'s.
(31) Obviously,    is increasing on   ; thus, by Lemma 7(i), {   ;  ≥ 1} is also a sequence of ND random variables.Taking   =  and  = () −1 in Lemma 8, for 1 <  1 <   , we obtain Replacing    by −   and by the same argument as above,