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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 218380, 15 pages
http://dx.doi.org/10.1155/2010/218380
Research Article

Convergence Properties for Asymptotically almost Negatively Associated Sequence

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

Received 20 July 2010; Revised 9 October 2010; Accepted 2 November 2010

Academic Editor: Ibrahim Yalcinkaya

Copyright © 2010 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

We get the strong law of large numbers, strong growth rate, and the integrability of supremum for the partial sums of asymptotically almost negatively associated sequence. In addition, the complete convergence for weighted sums of asymptotically almost negatively associated sequences is also studied.

1. Introduction

Definition 1.1. A finite collection of random variables is said to be negatively associated (NA) if, for every pair of disjoint subsets , of , whenever and are coordinate-wise nondecreasing such that this covariance exists. An infinite sequence is NA if every finite subcollection is NA.

The concept of negative association was introduced by Joag-Dev and Proschan [1] and Block et al. [2]. By inspecting the proof of maximal inequality for the NA random variables in Matuła [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) if there exists a nonnegative sequence as such that for all and for all coordinate-wise nondecreasing continuous functions and whenever the variances exist.

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. 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, and Wang et al. [7] established the law of the iterated logarithm for product sums. Recently, Yuan and An [8] established some Rosenthal-type inequalities for maximum partial sums of AANA sequence. As applications of these inequalities, they derived some results on convergence, where , and complete convergence. In addition, they estimated the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.

The main purpose of the paper is to study the strong law of large numbers, strong growth rate, and the integrability of supremum for AANA sequence. In addition, the complete convergence for weighted sums of AANA sequence is also studied.

Throughout the paper, we let be a sequence of AANA random variables defined on a fixed probability space . Denote . Let for some , and let be the indicator function of the set . For , let be the dual number of . We assume that is a positive increasing function on satisfying as and is the inverse function of . Since , it follows that . For easy notation, we let and . The denotes that there exists a positive constant such that . denotes a positive constant which may be different in various places. The main results of this paper are dependent on the following lemmas.

Lemma 1.3 (cf. Yuan and An [8, Lemma  2.1]). Let be a sequence of AANA random variables with mixing coefficients , and let be all nondecreasing (or nonincreasing) functions, then is still a sequence of AANA random variables with mixing coefficients .

Lemma 1.4. Let , and let be a sequence of AANA random variables with mixing coefficients and for each . If , then there exists a positive constant depending only on such that for all , where .

Proof. We use the same notations as that in the study by Yuan and An [8]. They proved that By (1.4) and Hölder's inequality, we have This completes the proof of the lemma.

We point out that Lemma 1.4 has been studied by Yuan and An [8]. But here we give the accurate coefficient . And Lemma 1.4 generalizes and improves the result of Lemma  2.2 in the study by Ko et al. [6].

Lemma 1.5 (cf. Fazekas and Klesov [9, Theorem  2.1] and Hu et al. [10, Lemma  1.5]). Let be a sequence of random variables. Let be a nondecreasing unbounded sequence of positive numbers, and let be nonnegative numbers. Let and be fixed positive numbers. Assume that, for each , then and with the growth rate where If further one assumes that for infinitely many , then

Lemma 1.6 (cf. Fazekas and Klesov [9, Corollary  2.1] and Hu [11, Corollary  2.1.1]). Let be a nondecreasing unbounded sequence of positive numbers, and let be nonnegative numbers. Denote for . Let be a fixed positive number satisfying (1.6). If then (1.8)–(1.11) hold.

Lemma 1.7 (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 1.8. Assume that the inverse function of satisfies If , then .

Proof. Since is an increasing function of , we have that The proof is complete.

2. Strong Law of Large Numbers and Growth Rate for AANA Sequence

Theorem 2.1. Let be a sequence of mean zero AANA random variables with , and let be a nondecreasing unbounded sequence of positive numbers; . Assume that then and with the growth rate where If further one assumes that for infinitely many , then

Proof. By Lemma 1.4, we have It follows from (2.1) that Thus, (2.2)–(2.5) follow from (2.6), (2.7), and Lemma 1.5 immediately. We complete the proof of the theorem.

Theorem 2.2. Let be a sequence of AANA random variables with , . Denote for and . Assume that then and with the growth rate where If further one assumes that for infinitely many , then In addition, for any ,

Proof. Assume that , , and . By Lemma 1.4, we can see that It is a simple fact that for all . It follows from (2.8) that That is to say that (1.12) holds. By Remark  2.1 in Fazekas and Klesov [9], (1.12) implies (1.13). By Lemma 1.6, we can obtain (2.9)–(2.14) immediately. By (2.13), it follows that The proof is complete.

Theorem 2.3. Let , where integer number , and let be a sequence of AANA random variables with for all and . Let be a nondecreasing unbounded sequence of positive numbers. Assume that then (1.8)–(1.11) hold (for ), where and is defined in Lemma 1.7.

Proof. Since , . By 's inequality, which implies that By Jensen's inequality, we have By (2.22)-(2.23) and 's inequality, It follows from Lemma 1.7 and (2.24) that It is a simple fact that where is a positive number depending only on and . By (2.19), The desired results follow from (2.25)–(2.27) and Lemma 1.5 immediately.

3. Complete Convergence for Weighted Sums of AANA Random Variables

Theorem 3.1. Let be a sequence of identically distributed AANA random variables with , , , and . Assume that the inverse function of satisfies (1.15). Let be a triangular array of positive constants such that
(i),
(ii) for some .
Then, for any ,

Proof. For each , denote It is easy to check that Therefore, Firstly, we will show that It follows from Lemma 1.8 and Kronecker's lemma that By , condition (i), (3.6), and , we can see that which implies (3.5). By (3.4) and (3.5), we can see that, for sufficiently large , To prove (3.1), it suffices to show that By Markov's inequality, Lemma 1.4, inequality, , and condition , we have It follows from that

Theorem 3.2. Let be a sequence of AANA random variables, and let be an array of positive numbers. Let be an increasing sequence of positive integers, and let be a sequence of positive numbers. If, for some , where integer number , , and for any , the following conditions are satisfied: and , then

Proof. Note that if the series is convergent, then (3.13) holds. Therefore, we will consider only such sequences for which the series is divergent.
Let Note that Using the inequality and Jensen's inequality, we can estimate in the following way: By (3.15), (3.16), and Lemma 1.7, we can get Therefore, we can conclude that (3.13) holds by (3.12) and (3.17).

Theorem 3.3. Let and let be a sequence of AANA random variables with and for . Let be an array of real numbers satisfying the condition and for some and , where integer number . Then, for any and ,

Proof. Take , , and in Theorem 3.2. By (3.18), we have following from . By the assumption for and (3.18), we get following from and . We get the desired result from Theorem 3.2 immediately. The proof is complete.

Theorem 3.4. Let be a sequence of AANA random variables satisfying , and let be an array of positive numbers. Let be an increasing sequence of positive integers, and let be a sequence of positive numbers. If, for some , , and for any , the following conditions are satisfied: then

Proof. The proof is similar to that of Theorem 3.2, so we omit it.

Acknowledgments

The authors are most grateful to the Editor Ibrahim Yalcinkaya and anonymous referee for careful reading of the manuscript and valuable suggestions, which helped to improve an earlier version of this paper. This paper was supported by the NNSF of China (Grant nos. 10871001, 61075009), 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), Academic innovation team of Anhui University (KJTD001B), and Natural Science Research Project of Suzhou College (2009yzk25).

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