Some Strong Convergence Theorems for Asymptotically almost Negatively Associated Random Variables

In this work, the complete moment convergence and Lp convergence for asymptotically almost negatively associated (AANA, in short) random variables are investigated. As an application, the complete convergence theorem for weighted sums of AANA random variables is obtained. These theorems obtained extend and improve some earlier results.


Introduction
Definition 1.1 A finite collection of random variables 1 2 , ,..., n X X X is said to be negatively associated ( f and 2 f are any real coordinatewise nondecreasing functions such that this covariance exists.
An infinite sequence { } ; 1 n X n ≥ of random variables is said to be NA if for every finite sub-collection is NA. The concept of NA was introduced by Joag-Dev and Proschan [4], and its probability limit properties have aroused wide interest because of their numerous applications in reliability theory, percolation theory and multivariate statistical analysis. By inspecting the proof of maximal inequality for NA random variables in Matula [7], one can also allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal [1,2] introduced the following dependence.  and for all coordinatewise nondecreasing continuous functions 1 f and 2 f whenever the variances exist.
The family of AANA sequence contains NA (in particular, independent) sequence (with ( ) 0, q n = 1 n ≥ ) and some more sequences of random variables which are not much deviated from being NA. Chandra and Ghosal [1] once pointed out that NA implies AANA, but AANA does not imply NA. Namely, AANA is much weaker than NA. Since, NA has been applied to the reliability theory, multivariate statistical analysis and percolation theory, and attracted extensive attentions. Hence, extending the limit properties of NA random variables to the wider case of AANA random variables is highly desirable in the theory and applications.
For recent various results and applications of AANA random variables, we can refer to that Chandra and Ghosal [1] obtained the Kolmogorov type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund; Chandra and Ghosal [2] established the almost sure convergence of weighted averages; Wang et al. [10] obtained the law of the iterated logarithm for product sums; Ko et al. [5] studied the Hájek-Rényi type inequality; Yuan and An [14] established some Rosenthal type inequalities; Yuan and Wu [15] studied the limiting behavior of the maximum of the partial sum under residual Cesàro alpha-integrability assumption; Wang et al. [11,12], Huang et al. [3] studied the complete convergence of weighted sums for arrays of rowwise AANA random variables and arrays of rowwise AANA random variables, respectively; Yang et al. [16] investigated the complete convergence of moving average process for AANA sequence; and Tang [9] studied the strong law of large numbers for general weighted sums, Shen and Wu [8] obtained some new complete convergence results and Feller-type weak law of large numbers, and so forth.
Recently, Liang and Su [6] obtained the following complete convergence result for weighted sums of NA random variables.
This paper is motivated by Liang and Su [6], Wang et al. [13]. We further study the convergence properties for AANA random variables and establish the complete moment convergence theorem and ( ) 1 2 p L p < < convergence theorem. As an application, the complete convergence and strong law of large numbers for weighted sums of AANA random variables are obtained. The obtained results extend and improve the above Theorem A and Theorem B.
The structure of this paper is as follows. In Section 2, some important lemmas are firstly provided and the complete moment convergence theorem for AANA random variables is presented. In Section 3, the p L convergence theorem is provided.
Throughout this paper, let { } A The symbol C will denote a positive constant which is not necessarily the same one in each appearance, n n a Cb ≤

Complete Moment Convergence
In order to prove our main results, the following lemmas are needed.
Then for 0, ≥ be an array of real numbers such that To prove (2.5), it needs only to show that 1 , It follows from (2.2) that Hence, the desired result of 1 I < ∞ follows from (2.10) immediately.
Secondly, we will show that 2 .
By the similar argument as the proof of (2.9), we can obtain that ( ) ( ) For convenience, let 1, Hence, the desired result of 2 I < ∞ follows from (2.2) and the above statements immediately. The proof of Theorem 2.1 is completed.   The proof of Theorem 2.2 is completed.

L p Convergence
In this section, we will state the L p convergence under some conditions.    Take into account the definition of i Z and (3.9), we can obtain that ( ) ( ) The proof of Corollary 3.2 is completed.