Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 863931, 24 pagesdoi:10.1155/2012/863931

Research Article

On Complete Convergence of Moving Average Process for AANA Sequence

Wenzhi Yang,¹ Xuejun Wang,¹ Nengxiang Ling,² and Shuhe Hu¹

¹School of Mathematical Science, Anhui University, Hefei 230039, China²School of Mathematics, Hefei University of Technology, Hefei 230009, China

Received 23 November 2011; Accepted 28 February 2012

Academic Editor: Cengiz Çinar

Copyright © 2012 Wenzhi Yang 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 investigate the moving average process such that , , where and is a sequence of asymptotically almost negatively associated (AANA) random variables. The complete convergence, complete moment convergence, and the existence of the moment of supermum of normed partial sums are presented for this moving average process.

1. Introduction

We assume that is a doubly infinite sequence of identically distributed random variables with . Let be an absolutely summable sequence of real numbers and be the moving average process based on the sequence . As usual, denotes the sequence of partial sums.

Under the assumption that is a sequence of independent identically distributed random variables, various results of the moving average process have been obtained. For example, Ibragimov [1] established the central limit theorem, Burton and Dehling [2] obtained a large deviation principle, and Li et al. [3] gave the complete convergence result for .

Many authors extended the complete convergence of moving average process to the case of dependent sequences, for example, Zhang [4] for -mixing sequence, Li and Zhang [5] for NA sequence. The following Theorems A and B are due to Zhang [4] and Kim et al. [6], respectively.

Theorem A. Suppose that is a sequence of identically distributed -mixing random variables with and is as in (1.1). Let be a slowly varying function and , . If and , then

Theorem B. Suppose that is a sequence of identically distributed -mixing random variables with and and is as in (1.1). Let be a slowly varying function and . If , then where .

Chen and Gan [7] investigated the moments of maximum of normed partial sums of -mixing random variables and gave the following result.

Theorem C. Let and . Assume that is a mean zero sequence of identically distributed -mixing random variables with the maximal correlation coefficient rate , where if and if . Denote . Then are all equivalent.

Chen et al. [8] and Zhou [9] also studied limit behavior of moving average process under -mixing assumption. For more related details of complete convergence, one can refer to Hsu and Robbins [10], Chow [11], Shao [12], Li et al. [3], Zhang [4], Li and Zhang [5], Chen and Gan [7], Kim et al. [6], Sung [13–15], Chen and Li [16], Zhou and Lin [17], and so forth.

Inspired by Zhang [4], Kim et al. [6], Chen and Gan [7], Sung [13–15], and other papers above, we investigate the limit behavior of moving average process under AANA sequence, which is weaker than NA and obtain some similar results of Theorems A, B, and C. The main results can be seen in Section 2 and their proofs are given in Section 3.

Recall that the sequence is stochastically dominated by a nonnegative random variable if

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 coordinatewise nondecreasing such that this covariance exists.

An infinite sequence is NA if every finite subcollection is NA.

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 concept of NA sequence was introduced by Joag-Dev and Proschan [18]. For the basic properties and inequalities of NA sequence, one can refer to Joag-Dev and Proschan [18] and Matula [19]. The family of AANA sequence contains NA (in particular, independent) sequence (with ) and some more sequences of random variables which are not much deviated from being negatively associated. An example of an AANA which is not NA was constructed by Chandra and Ghosal [20, 21]. For various results and applications of AANA random variables can be found in Chandra Ghosal [21], Wang et al. [22], Ko et al. [23], Yuan and An [24], and Wang et al. [25, 26] among others.

For simplicity, in this paper we consider the moving average process: where and is a mean zero sequence of AANA random variables.

The following lemmas are our basic techniques to prove our results.

Lemma 1.3. Let be a sequence of AANA random variables with mixing coefficients . If are all nondecreasing (or nonincreasing) continuous functions, then is still a sequence of AANA random variables with mixing coefficients .

Remark 1.4. Lemma 1.3 comes from Lemma 2.1 of Yuan and An [24], but the functions of in Lemma 2.1 of Yuan and An [24] are written to be all nondecreasing (or nonincreasing) functions. According to the definition of AANA, should be all nondecreasing (or nonincreasing) continuous functions.

Lemma 1.5 (cf. Wang et al. [25, Lemma 1.4]). Let and be a mean zero sequence of AANA random variables with mixing coefficients . If , then there exists a positive constant depending only on such that for all , where .

Lemma 1.6 (cf. Wu [27, Lemma 4.1.6]). Let be a sequence of random variables, which is stochastically dominated by a nonnegative random variable . For any and , the following two statements hold: where and are positive constants.

Throughout the paper, is the indicator function of set , and denote some positive constants not depending on , which may be different in various places.

2. The Main Results

Theorem 2.1. Let , and . Assume that is a moving average process defined in (1.8), where is a mean zero sequence of AANA random variables with and stochastically dominated by a nonnegative random variable . If , then for every ,

Theorem 2.2. Let the conditions of Theorem 2.1 hold. Then for every ,

Theorem 2.3. Let and . Assume that is a moving average process defined in (1.8), where is a mean zero sequence of AANA random variables with and stochastically dominated by a nonnegative random variable with . Suppose that Then

3. The Proofs of Main Results

Proof of Theorem 2.1. Firstly, we show that the moving average process (1.8) converges almost surely under the conditions of Theorem 2.1. Since , it has , following from the condition . On the other hand, by Lemma 1.6 with and , one has Consequently, by the condition , we have that which implies converges almost surely.Note that Since and , one has . Combining with , and Lemma 1.6, we can find that Meanwhile, Let Hence, for for all , there exists an such that Denote Noting that , we can find For , by Markov's inequality, , Lemma 1.6 and , it has Since is a nondecreasing continuous function of , we can find by using Lemma 1.3 that is a mean zero AANA sequence and , , . Consequently, by the property of AANA, Markov's inequality, Hölder's inequality, Lemma 1.5, inequality, and Lemma 1.6, we can check that Since , it can be seen by that By the proof of (3.10), we have . Therefore, (2.1) follows from (3.9), (3.10), (3.11), and (3.12).Inspired by the proof of Theorem 12.1 of Gut [28], it can be checked that Combining (2.1) with the inequality above, we obtain (2.2) immediately.

Proof of Theorem 2.2. For all , it has By Theorem 2.1, in order to prove (2.3), we only have to prove that For , let Since , , it is easy to see that For , by Markov's inequality, , Lemma 1.6 and , we get that From the fact that is a mean zero AANA sequence and , , similar to the proof of (3.11), we have It follows from and that By the proof of (3.18), one has On the other hand, by the property , we have Thus, by Lemma 1.6 and the proof of (3.18), it can be seen that Consequently, by (3.14), (3.17), (3.18), (3.19), (3.20), (3.21), and (3.23) and Theorem 2.1, (2.3) holds true.Next, we prove (2.4). It is easy to see that Therefore, (2.4) holds true following from (2.3).

Proof of Theorem 2.3. Similar to the proof of Theorem 2.1, by and , converges almost surely. It can be seen that For , let Since , then For , similar to (3.18), by Markov's inequality and Lemma 1.6, one has For the case , if , then If , then Otherwise for , it has Similarly, for the case , if , then If , then Otherwise, for , it follows On the other hand, for the case , if , then If , then For , it has Consequently, by (3.28), the conditions of Theorem 2.3 and inequalities above, we obtain that Since is a mean zero AANA sequence and , , , similar to the proof of (3.11), we obtain that Similar to the proof of Theorem 1.1 of Chen and Gan [7], by and the conditions of Theorem 2.3, we have that On the other hand, by the proof of (3.28) and (3.38), it follows Similar to the proof of (3.23), by the property and the proofs of (3.28) and (3.38), one has Consequently, by (3.25), (3.27), (3.28), (3.38), (3.39), (3.40), (3.41), and (3.42), we finally obtain (2.6).

Remark 3.1. Zhou and Lin [17] obtained the result (2.6) for partial sums of moving average process under -mixing sequence. But there is one problem in their proof. On page 694 of Zhou and Lin [17], they presented that where and . For the case , by taking , we cannot get that Here, we give a counter example to illustrate this problem. Assume that the density function of nonnegative random variable is Obviously, it can be found that But for ,

Acknowledgments

The authors are deeply grateful to Editor Cengiz Çinar and two anonymous referees for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work is supported by the NNSF of China (11171001), HSSPF of the Ministry of Education of China (10YJA910005), Mathematical Tianyuan Youth Fund (11126176), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Natural Science Foundation of Anhui Province (1208085QA03), and Academic Innovation Team of Anhui University (KJTD001B).

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