Complete Moment Convergence for Negatively Dependent Sequences of Random Variables
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.
1. Introduction and Main Results
Random variables and are said to be negative quadrant dependent (NQD) iffor all 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) impliesfor all . Moreover, it follows that (2) implies (1) and, hence, (1) and (2) are equivalent. However, Ebrahimi and Ghosh  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.
Definition 1. Random variables are said to be negatively dependent (ND) if, for all real , An infinite sequence of random variables is said to be ND if every finite subset is ND.
Definition 2. Random variables , are said to be negatively associated (NA) if, for every pair of disjoint subsets and of , where and are increasing for every variable (or decreasing for every variable) function such that this covariance exists. A sequence of random variables is said to be NA if its every finite subfamily is NA.
The definition of PNQD is given by Lehmann . The definition of NA is introduced by Joag-Dev and Proschan , and the concept of ND is given by Bozorgnia et al. . 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 in Wu and Jiang  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 well-known 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. , Fakoor and Azarnoosh , Asadian et al. , Wu [5, 8], Wang et al. , and Liu et al. ). 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  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 , Gut and Stadtmüller , Sung , Guo , and Qiu and Chen [16, 17]. In addition, Qiu and Chen  obtained complete moment convergence theorems for independent identically distributed sequences of random variables with , , . A natural question is whether there is any type of complete moment convergence theorems for . In this paper, 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.
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 such that for sufficiently large , mean , and denotes an indicator function.
Theorem 3. Let , be a sequence of ND identically distributed random variables with partial sums , . Suppose thatand thenConversely, if (6) holds for and some , then
For , we have the following.
Theorem 4. Let be a sequence of ND identically distributed random variables with partial sums , . Suppose thatand thenConversely, if for some , then
The following four lemmas play important roles in the proof of our theorems.
Lemma 7 (Bozorgnia et al. ). Let be a sequence of ND random variables.(i)Let be a sequence of Borel functions; all of them are monotone increasing (or all are monotone decreasing). Then, is a sequence of ND r.v.’s.(ii)Let be nonnegative. Then, In particular, let be all nonnegative (or nonpositive) real numbers. Then,
Lemma 8. Let be a sequence of ND identically distributed random variables with for any . Assume that is a sequence of positive real numbers such that as . Then, for , , and a positive integer ,
Proof. Obviously, from condition for any . By Lemma 7(ii), () in Wang et al.  holds for any . Together with condition , we know that the conditions of Theorem in Wang et al.  are satisfied. Therefore, by Theorem in Wang et al. , for any ,
Lemma 9. For any random variable ,
Proof. Let denote that there exist constants and such that for sufficiently large . We have and it follows that (17) holds.
Note that and hence, using similar methods used to prove (17), we can prove that (18) holds.
Lemma 10. Let be a sequence of ND random variables. Then, for any , there exists a positive constant such that, for all ,
Further, if as , then there exists a positive constant such that, for all ,
Proof of Theorem 3. Note thatHence, in order to establish (6), it suffices to prove, for any ,Firstly, we prove (24). Let be arbitrary; define, for , It is easy to getFrom (5) and the Markov inequality,Set ; then ; using the obvious inequality , it follows that, for any ,Let and ; then by inequality and ,By (5), , and , therefore, by combination with (29) and (30), we getObviously, is increasing on ; thus, by Lemma 7(i), is also a sequence of ND random variables. Taking and in Lemma 8, for , we obtainReplacing by and by the same argument as above, also holds. Hence,Thus, by combination with (27) and (28) and the fact that for all , we obtain that, for , Hence, (24) holds. Next, we prove (25).
Let . Replace by in and . Using similar methods to those used in the proof of (27)–(34), there is such thatThus, and it implies that That is, (25) holds.
Conversely, if (6) holds for and some , then, by , we have by combination with , it follows thatand it implies that , . Thus, by Lemma 10, there is such that Consequently, by (40), and, hence, from Lemma 9. This completes the proof of Theorem 3.
Proof of Theorem 4. Let and be defined as Theorem 3. Let , and ; then, from and being monotonically increasing on .
Thus, by (8), ; therefore, similar to (29), we have By Lemma 8, for , we getReplacing by and by the same argument as above, also holds. Hence, On the other hand, from (8) and the Markov inequality, Hence, together with (27),Similar to (36) and the above discussion, for , Thus, and it implies that Thus, by combination with (23) and (49), (9) holds.
Conversely, by (18), using similar methods to those used in the proof of (7), we can get (10). This completes the proof of Theorem 4.
The authors declare that they have no competing interests.
Qunying Wu conceived the study and drafted, completed, read, and approved the final paper. Yuanying Jiang conceived the study and completed, read, and approved the final paper.
This work was supported by the National Natural Science Foundation of China (11361019) and the Support Program of the Guangxi China Science Foundation (2015GXNSFAA139008).
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