Complete Moment Convergence and Mean Convergence for Arrays of Rowwise Extended Negatively Dependent Random Variables

The authors first present a Rosenthal inequality for sequence of extended negatively dependent (END) random variables. By means of the Rosenthal inequality, the authors obtain some complete moment convergence and mean convergence results for arrays of rowwise END random variables. The results in this paper extend and improve the corresponding theorems by Hu and Taylor (1997).


Introduction
The concept of the complete convergence was introduced by Hsu and Robbins [1]. A sequence of random variables { , ≥ 1} is said to converge completely to a constant if In view of the Borel-Cantelli lemma, the above result implies that → almost surely. Therefore, the complete convergence is a very important tool in establishing almost sure convergence of summation of random variables as well as weighted sums of random variables.
Chow [2] presented the following more general concept of the complete moment convergence. Let { , ≥ 1} be a sequence of random variables and > 0, > 0, and > 0. If − } + < ∞ for some or all > 0, (2) then the above result was called the complete moment convergence.
The following concept of negatively orthant dependent (NOD) random variables was introduced by Ebrahimi and Ghosh [3].
Random variables 1 , . . . , are said to be NOD if they are both NUOD and NLOD.
Liu [4] extended the above negatively dependent structure and introduced the concept of extended negatively dependent (END) random variables.

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The Scientific World Journal As described in Liu [4], the END structure is substantially more comprehensive than the NOD structure in that it can reflect not only a negative dependence structure but also a positive one, to some extent. Joag-Dev and Proschan [5] also pointed out that negatively associated (NA) random variables must be NOD and NOD is not necessarily NA. Since NOD implies END, NA random variables are END.
The convergence properties of NOD random sequences were studied in the different aspects. We refer reader to Taylor et al. [6] and Ko et al. [7,8] for the almost sure convergence; Wu et al. [9] for the weak convergence and 1 -convergence; Amini and Bozorgnia [10], Gan and Chen [11], Wu [12], Wu and Zhu [13], Qiu et al. [14], and Shen [15] for complete convergence; and Wu and Zhu [13] and Wu et al. [9] for complete moment convergence.
Since the paper of Liu [4] appeared, the probabilistic properties for END random variables have been studied by Chen et al. [16], Wu and Guan [17], and Qiu et al. [18]. Since NOD implies END and a great numbers of articles for NOD random variables have appeared in literature, it is very interesting to investigate convergence properties of this wider END class.
For a triangular array of rowwise independent random variables { , 1 ≤ ≤ , ≥ 1}, we let { , ≥ 1} be a sequence of positive real numbers with ↑ ∞, and {Ψ( )} be a positive, even function such that for some nonnegative integer . Conditions are given as where is a positive integer. Hu and Taylor [19] proved the following theorems. Sung [20], Gan and Chen [21], and Wu and Zhu [13] extended Theorems A and B to the cases of -valued random elements, NA random variables, and NOD random variables, respectively. The goal of this paper is to study complete moment convergence and mean convergence for arrays of rowwise END random variables.
In this work, the authors first present a Rosenthal inequality for sequence of END random variables. By means of the Rosenthal inequality, the authors obtain the complete moment convergence result for arrays of rowwise END random variables, which extends and improves Theorems A and B. In addition, the authors study mean convergence for arrays of rowwise END random variables which was not considered by Hu and Taylor [19].
Throughout this paper, the symbol represents positive constants whose values may change from one place to another.
be an array of rowwise END random variables, and let { , ≥ 1} be a sequence of positive real numbers with ↑ ∞. Also, let {Ψ( )} be a positive, even function satisfying (11) for 1 ≤ < .
The Scientific World Journal 3 Remark 5. Since an independent random variable sequence is a special END sequence, Theorems 3 and 4 hold for arrays of rowwise independent random variables. Note that implies (10). Therefore, the conclusion of Theorem 3 is stronger than those of Theorems A and B.

Proofs
To prove our main results, we need the following lemmas.
where is a positive constant depending only on .
Proof of Theorem 4. Following the notations of the proof in Theorem 3. To start with, we prove (15) for the case 1 < ≤ 2.
For all > 0, Without loss of generality we may assume 0 < < 1. By Markov inequality, (11), and (14), we have From (11), (7), and (14), we have Therefore, while is sufficiently large, for ≥ , we have (31). Let = [ ] + 1; by (31), Lemma 7, and inequality, we have By similar argument as in the proof of 41 < ∞, we can prove Therefore, by similar argument as in the proof of 42 < ∞, we can prove By similar argument as in the proof of 5 → 0, we can prove The proof of (15) for the case > 2 is similar to that of (ii) in Theorem 3, so we omit the details. The proof is complete.