%0 Journal Article
%T On Complete Moment Convergence of Weighted Sums for Arrays of Rowwise Negatively Associated Random Variables
%A Mingle Guo
%A Dongjin Zhu
%J Journal of Probability and Statistics
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/850608
%X The complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables is investigated. Some sufficient conditions for complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables are established. Moreover, the results of Baek et al. (2008), are complemented. As an application, the complete moment convergence of moving average processes based on a negatively associated random sequences is obtained, which improves the result of Li et al. (2004). 1. Introduction Let be a sequence of random variables and, as usual, set . When are independent and identically distributed (i.i.d.), Baum and Katz [1] proved the following remarkable result concerning the convergence rate of the tail probabilities for any . Theorem A (see [1]). Let and . Then if and only if , where whenever . There is an interesting and substantial literature of investigation apropos of extending the Baum-Katz theorems along a variety of different paths. One of these extensions is due to Chow [2] who established the following refinement which is a complete moment convergence result for sums of i.i.d. random variables. Theorem B (see [2]). Let , , and . Suppose that . Then Recently, Baum-Katz theorem is extended to the case of dependence random variables. Liang [3] obtained some general results on the complete convergence of weighted sums of negatively associated random variables. Li and Zhang [4] showed complete moment convergence for moving average processes under negative association as follows. Theorem C (see [4]). Suppose that , where is a sequence of real numbers with and is a sequence of identically distributed and negatively associated random variables with . Let be a slowly varying function and . Then implies that Kuczmaszewska [5] proposed a very general result for complete convergence of rowwise negatively associated arrays of random variables which is stated in Theorem D. Theorem D (see [5]). Let be an array of rowwise negatively associated random variables and let be an array of real numbers. Let be an increasing sequence of positive integers and let be a sequence of positive real numbers. If for some and any the following conditions are fulfilled:(a) ,(b) ,(c) ,then Baek et al. [6] discussed complete convergence of weighted sums for arrays of rowwise negatively associated random variables and obtained the following results. Theorem E (see [6]). Let be an array of rowwise negatively associated random variables with and for all and . Suppose that , and that is an array of constants such that (a)If and
%U http://www.hindawi.com/journals/jps/2012/850608/