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Journal of Probability and Statistics
Volume 2012 (2012), Article ID 850608, 12 pages
http://dx.doi.org/10.1155/2012/850608
Research Article

On Complete Moment Convergence of Weighted Sums for Arrays of Rowwise Negatively Associated Random Variables

School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China

Received 8 May 2012; Revised 29 July 2012; Accepted 13 August 2012

Academic Editor: Shein-chung Chow

Copyright © 2012 Mingle Guo and Dongjin Zhu. 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

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 there exists some such that , and , then, under , one has (b)If , then, under , (1.7) remain true.

In this paper, the authors take the inspiration in [5, 6] and discuss the complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables by applying truncation methods, which extend the results of [5, 6]. As an application, the complete moment convergence of moving average processes based on a negatively associated random sequences is obtained, which extend the result of Li and Zhang [4].

For the proof of the main results, we need to restate a few definitions and lemmas for easy reference. Throughout this paper, will represent positive constants whom their value may change from one place to another. The symbol denotes the indicator function of , indicate the maximum integer not larger than . For a finite set , the symbol denotes the number of elements in the set .

Definition 1.1. A finite family of random variables is said to be negatively associated (abbreviated to NA in the following), if for every pair disjoint subsets and of and any real nondecreasing coordinate-wise functions on and on whenever the covariance exists.
An infinite family of random variables is NA if every finite subfamily is NA.

The definition of negatively associated was introduced by Alam and Saxena [7] and was studied by Joag-Dev and Proschan [8] and Block et al. [9]. As pointed out and proved by Joag-Dev and Proschan, a number of well-known multivariate distributions possess the NA property. Negative association has found important and wide applications in multivariate statistical analysis and reliability. Many investigators discuss applications of negative association to probability, stochastic processes, and statistics.

Definition 1.2. A sequence of random variables is said to be stochastically dominated by a random variable (write ) if there exists a constant , such that for all and .

The following lemma is a well-known result.

Lemma 1.3. Let the sequence of random variables be stochastically dominated by a random variable . Then for any

Definition 1.4. A real-valued function , positive and measurable on for some , is said to be slowly varying if for each .

By the properties of slowly varying function, we can easily prove the following two lemmas. Here we omit the details of the proof.

Lemma 1.5. Let be a slowly varying function as .(i) for any and positive integer .(ii) for any and positive integer .

Lemma 1.6. Let be a random variable and let be a slowly varying function as .(i)If , then if and only if .(ii)If , then if and only if .

The following lemma will play an important role in the proof of our main results. The proof is according to Shao [10].

Lemma 1.7. Let be a sequence of NA random variables with mean zero and for every . Then

By monotone convergence and (1.10), we have the following lemma.

Lemma 1.8. Let be a sequence of NA random variables with mean zero and for every . Then

Using Lemma 1.4, Lemma 1.5, and Theorem 2.11 in Sung [11], we obtain the following lemmas.

Lemma 1.9. Let be an array of rowwise NA random variables with for . Let be a sequence of real numbers. If for some , the following conditions are fulfilled:(a);(b);(c).
Then

Lemma 1.10. Let be an array of rowwise NA random variables with for . Let be a sequence of real numbers. If for some , the following conditions are fulfilled:(a);(b);(c).
Then

2. Main Results

Now we state our main results. The proofs will be given in Section 3.

Theorem 2.1. Let be an array of rowwise NA random variables with and stochastically dominated by a random variable . Suppose that is a slowing varying function and that is an array of constants such that (i)If and there exists some such that , and , then implies (ii)If , assume also for all . Then implies

Theorem 2.2. Let be an array of rowwise NA random variables with and stochastically dominated by a random variable . Suppose that is a slowing varying function and that is an array of constants such that (i)If and there exists some such that , and . Then implies (ii)If , assume also for all . Then implies

Remark 2.3. If (2.7) and (2.8) hold, then for all , we have Thus, we improve the results of Baek et al. [6] to supreme value of partial sums.

Remark 2.4. If , then implies that (2.10) holds. In fact,

Corollary 2.5. Under the conditions of Theorem 2.2,

Corollary 2.6. Let be an array of rowwise NA random variables with and stochastically dominated by a random variable . Suppose that is a slowing varying function.(1)Let and . If , then (2)Let . If , then

Theorem 2.7. Suppose that , where is a sequence of real numbers with , and is a NA random sequence with and is stochastically dominated by a random variable . Let be a slowly varying function.(1)Let . If , then (2)Let . If , then

Remark 2.8. Theorem 2.7 obtains the result about the complete moment convergence of moving average processes based on an NA random sequence with different distributions. The result of Li and Zhang [4] is a special case of Theorem 2.7 (1). Moreover, our result covers the case of , which was not considered by Li and Zhang.

3. Proofs of the Main Results

Proof of Theorem 2.1. Since , where and , we have So, without loss of generality, we can assume . From (2.1) and (2.2), without loss of generality, we assume Put in Lemma 1.9. Noting that , by Lemma 1.3 and Lemma 1.7, we have Since , we can take some such that . Observe that Hence, choosing large enough such that , we have By (3.3) and Lemma 1.3, we have Set . Then . Note also that for all , Hence, we have Note that Choosing large enough such that , we obtain by Lemma 1.6 and (3.8) that Noting that , by (3.8) and Lemma 1.5, we see From (3.6), (3.9), (3.10), and (3.11), we know that By (3.3), (3.5), and (3.12), we see that (a), (b), and (c) in Lemma 1.9 with replaced by are fulfilled. Since is also an array of rowwise NA random variables, by Lemma 1.9, we complete the proof of (2.3).
Next, we prove (2.4). If , then . Similarly for the proof of (3.3), noting that , we have Taking , from the proof of (3.9), (3.10), and (3.11), we obtain Thus, for , (a), (b), and (c) in Lemma 1.9 with replaced by are fulfilled. So (2.4) holds.

Proof of Theorem 2.2. By Lemma 1.10, the rest of the proof is similar to that of Theorem 2.1 and is omitted.

Proof of Corollary 2.5. Note that Therefore, by (2.7) and (2.8), we prove that (2.12) holds.

Proof of Corollary 2.6. By applying Theorem 2.1, taking for , and for , then we obtain (2.13). Similarly, taking for , and for , we obtain (2.14) by Theorem 2.1.

Proof of Theorem 2.7. Let and for all . Since , we have and . By applying Corollary 2.5, taking , we obtain Therefore, (2.15) and (2.16) hold.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Grant no. 10901003), the Key Project of Chinese Ministry of Education (Grant no. 211077), and the Anhui Provincial Natural Science Foundation (Grant no. 10040606Q30).

References

  1. L. E. Baum and M. Katz, “Convergence rates in the law of large numbers,” Transactions of the American Mathematical Society, vol. 120, pp. 108–123, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. Y. S. Chow, “On the rate of moment convergence of sample sums and extremes,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 16, no. 3, pp. 177–201, 1988. View at Zentralblatt MATH
  3. H.-Y. Liang, “Complete convergence for weighted sums of negatively associated random variables,” Statistics & Probability Letters, vol. 48, no. 4, pp. 317–325, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. L. Yun-Xia and Z. Li-Xin, “Complete moment convergence of moving-average processes under dependence assumptions,” Statistics & Probability Letters, vol. 70, no. 3, pp. 191–197, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. A. Kuczmaszewska, “On complete convergence for arrays of rowwise negatively associated random variables,” Statistics & Probability Letters, vol. 79, no. 1, pp. 116–124, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J.-I. Baek, I.-B. Choi, and S.-L. Niu, “On the complete convergence of weighted sums for arrays of negatively associated variables,” Journal of the Korean Statistical Society, vol. 37, no. 1, pp. 73–80, 2008. View at Publisher · View at Google Scholar
  7. K. Alam and K. M. L. Saxena, “Positive dependence in multivariate distributions,” Communications in Statistics A, vol. 10, no. 12, pp. 1183–1196, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. K. Joag-Dev and F. Proschan, “Negative association of random variables, with applications,” The Annals of Statistics, vol. 11, no. 1, pp. 286–295, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. H. W. Block, T. H. Savits, and M. Shaked, “Some concepts of negative dependence,” The Annals of Probability, vol. 10, no. 3, pp. 765–772, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Q.-M. Shao, “A comparison theorem on moment inequalities between negatively associated and independent random variables,” Journal of Theoretical Probability, vol. 13, no. 2, pp. 343–356, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. S. H. Sung, “Moment inequalities and complete moment convergence,” Journal of Inequalities and Applications, vol. 2009, Article ID 271265, 14 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH