Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 717126, 11 pagesdoi:10.1155/2011/717126

Research Article

Complete Convergence for Arrays of Rowwise Asymptotically Almost Negatively Associated Random Variables

Xuejun Wang, Shuhe Hu, and Wenzhi Yang

School of Mathematical Science, Anhui University, Hefei 230039, China

Received 24 April 2011; Revised 3 September 2011; Accepted 18 September 2011

Academic Editor: Wei-Der Chang

Copyright © 2011 Xuejun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be an array of rowwise asymptotically almost negatively associated random variables. Some sufficient conditions for complete convergence for arrays of rowwise asymptotically almost negatively associated random variables are presented without assumptions of identical distribution. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of asymptotically almost negatively associated random variables is obtained.

1. Introduction

The concept of complete convergence was introduced by Hsu and Robbins [1] as follows. A sequence of random variables is said to converge completely to a constant if for all . In view of the Borel-Cantelli lemma, this implies that almost surely (a.s.). The converse is true if the are independent. Hsu and Robbins [1] proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Since then many authors studied the complete convergence for partial sums and weighted sums of random variables. The main purpose of the present investigation is to provide the complete convergence results for weighted sums of asymptotically almost negatively associated random variables and arrays of rowwise asymptotically almost negatively associated random variables.

Firstly, let us recall the definitions of negatively associated and asymptotically almost negatively associated random variables.

Definition 1.1. A finite collection of random variables is said to be negatively associated (NA, in short) if for every pair of disjoint subsets of , whenever and are coordinatewise nondecreasing such that this covariance exists. An infinite sequence is NA if every finite subcollection is negatively associated.An array of random variables is called rowwise NA random variables if, for every , is a sequence of NA random variables.

The concept of negative association was introduced by Joag-Dev and Proschan [2]. By inspecting the proof of maximal inequality for the NA random variables in Matula [3], one also can allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal [4, 5] introduced the following dependence.

Definition 1.2. A sequence of random variables is called asymptotically almost negatively associated (AANA, in short) if there exists a nonnegative sequence as such that for all and for all coordinatewise nondecreasing continuous functions and whenever the variances exist.An array of random variables is called rowwise AANA random variables if, for every is a sequence of AANA random variables.

The family of AANA sequence contains NA (in particular, independent) sequences (with ) and some more sequences of random variables which are not much deviated from being negatively associated. An example of an AANA sequence which is not NA was constructed by Chandra and Ghosal [4].

Since the concept of AANA sequence was introduced by Chandra and Ghosal [4], many applications have been found. See, for example, Chandra and Ghosal [4] derived the Kolmogorov type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund, Chandra and Ghosal [5] obtained the almost sure convergence of weighted averages, Ko et al. [6] studied the Hájek-Rényi type inequality, Wang et al. [7] established the law of the iterated logarithm for product sums, Yuan and An [8] established some Rosenthal type inequalities for maximum partial sums of AANA sequence, and Wang et al. [9] obtained some strong growth rate and the integrability of supremum for the partial sums of AANA random variables, and so forth.

Our goal in this paper is to study the complete convergence for arrays of rowwise AANA random variables under some moment conditions. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of AANA random variables is obtained. We will give some sufficient conditions for complete convergence for an array of rowwise AANA random variables without assumption of identical distribution. The results presented in this paper are obtained by using the truncated method and the classical maximal type inequality of AANA random variables (Lemma 1.5 below).

Throughout the paper, let be an array of rowwise AANA random variables with the mixing coefficients in each row. For , let be the dual number of . Let be the indicator function of the set . denotes a positive constant which may be different in various places and stands for .

Definition 1.3. An array of random variables is said to be stochastically dominated by a random variable if there exists a positive constant such that for all , and .

The following lemmas are useful for the proofs of the main results.

Lemma 1.4 (cf. Yuan and An [8, Lemma 2.1]). Let be a sequence of AANA random variables with mixing coefficients , let be all nondecreasing (or all nonincreasing) functions, then is still a sequence of AANA random variables with mixing coefficients .

Lemma 1.5 (cf. Yuan and An [8, Theorem 2.1]). Let be a sequence of AANA random variables with for all and , where integer number . If , then there exists a positive constant depending only on such that for all ,

Lemma 1.6. Let be a sequence of random variables which is stochastically dominated by a random variable . For any and , the following two statements hold: where and are positive constants.

2. Main Results

Let be an array of rowwise AANA random variables with the mixing coefficients in each row, and let be an array of real numbers. Let be a sequence of AANA random variables with the mixing coefficients and let be a sequence of real numbers. We consider the following conditions. There exist some with and some with such that , and assume further that if .There exists some such that , where integer number . For some and , There exist some with and some with such that , and assume further that if .There exists some with such that and assume further that if .There exists some with such that and assume further that if .

Our main results are as follows.

Theorem 2.1. Let be an array of rowwise AANA random variables which is stochastically dominated by a random variable , and let be an array of real numbers. Suppose that the conditions (H₁)–(H₃) are satisfied. Then, for any , where and .

Proof. For fixed , define It is easy to check that for any , which implies that Firstly, we will show that By and Hölder's inequality, we have for that Hence, when , we have by , (1.6) of Lemma 1.6, (2.7) (taking ), Markov's inequality, and (2.1) that Elementary Jensen's inequality implies that for any , Therefore, when , we have by (1.5) of Lemma 1.6, (2.9), Markov's inequality, and (2.1) that By (2.8) and (2.10), we can get (2.6) immediately. Hence, for large enough, To prove (2.2), we only need to show that By Definition 1.3, Markov's inequality and (2.1), we can see that For fixed , it is easily seen that are still AANA random variables by Lemma 1.4. For , it follows from Lemma 1.5, 's inequality, and Jensen's inequality that Taking , which implies that . It follows from 's inequality, (1.5) of Lemma 1.6, (2.9), Markov's inequality, and (2.1) that By 's inequality, (1.5) of Lemma 1.6, (2.9), and Jensen's inequality, we can get that Therefore, the desired result (2.2) follows from (2.13)–(2.16) immediately. This completes the proof of the theorem.

Similar to the proof of Theorem 2.1, we can get the following result for sequences of AANA random variables.

Theorem 2.2. Let be a sequence of AANA random variables which is stochastically dominated by a random variable , and let be an array of real numbers. Suppose that the conditions (H₁)–(H₃) are satisfied ( is replaced by in ). Then, for any , where and .

The following result provides the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of AANA sequence of random variables.

Theorem 2.3. Let be a sequence of AANA random variables which is stochastically dominated by a random variable and be a sequence of real numbers. Suppose that the conditions (H₂)–(H₄) are satisfied. Then for any , where and for .

Proof. Similar to the proof of Theorem 2.1, we can get (2.18) immediately, which yields that Therefore, By Borel-Cantelli lemma, we obtain that For all positive integers , there exists a positive integer such that . We have by (2.22) that which implies (2.19). This completes the proof of the theorem.

Remark 2.4. In Theorems 2.1–2.3, the condition (H₁) or (H₄) is needed. Under the weaker condition ((H₅) or (H₆)) than ((H₁) or (H₄)), we can get the following Theorems 2.5–2.7. The details of their proofs are omitted.

Theorem 2.5. Let be an array of rowwise AANA random variables which is stochastically dominated by a random variable , and let be an array of real numbers. Suppose that the conditions (H₂), (H₃), and (H₅) are satisfied. Then, for any , where .

Theorem 2.6. Let be a sequence of AANA random variables which is stochastically dominated by a random variable , and let be an array of real numbers. Suppose that the conditions (H₂), (H₃) and (H₅) are satisfied ( is replaced by in (H₅)). Then, for any , where .

Theorem 2.7. Let be a sequence of AANA random variables which is stochastically dominated by a random variable , and let be a sequence of real numbers. Suppose that the conditions (H₂), (H₃), and (H₆) are satisfied. Then, for any , where and for .

Acknowledgments

The authors are most grateful to the Editor Wei-Der Chang and anonymous referees for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the NNSF of China (11171001), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Talents Youth Fund of Anhui Province Universities (2010SQRL016ZD), Youth Science Research Fund of Anhui University (2009QN011A), and the academic innovation team of Anhui University (KJTD001B).

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