`Journal of Probability and StatisticsVolume 2011, Article ID 708087, 10 pageshttp://dx.doi.org/10.1155/2011/708087`
Research Article

Strong Laws of Large Numbers for Arrays of Rowwise NA and LNQD Random Variables

College of Science, Guilin University of Technology, Guilin 541004, China

Received 25 May 2011; Revised 21 October 2011; Accepted 23 October 2011

Copyright © 2011 Jiangfeng Wang and Qunying Wu. 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

Some strong laws of large numbers and strong convergence properties for arrays of rowwise negatively associated and linearly negative quadrant dependent random variables are obtained. The results obtained not only generalize the result of Hu and Taylor to negatively associated and linearly negative quadrant dependent random variables, but also improve it.

1. Introduction

Let be a sequence of independent distributed random variables. The Marcinkiewicz-Zygmund strong law of large numbers (SLLN) provides that if and only if . The case is due to Kolmogorov. In the case of independence (but not necessarily identically distributed), Hu and Taylor [1] proved the following strong law of large numbers.

Theorem 1.1. Let be a triangular array of rowwise independent random variables. Let be a sequence of positive real numbers such that . Let be a positive, even function such that is an increasing function of and is a decreasing function of , respectively, that is, for some positive integer . If and where is a positive integer, then

Definition 1.2 (cf. [2]). A finite family of random variables is said to be negatively associated (NA, in short) if, for any disjoint subsets and of and any real coordinate-wise nondecreasing functions on on and on , whenever the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA. This concept was introduced by Joag-Dev and Proschan [2].

Definition 1.3 (cf. [3, 4]). Two random variables and are said to be negative quadrant dependent (NQD, in short) if, for any , A sequence of random variables is said to be pairwise NQD if and are NQD for all and .

Definition 1.4 (cf. [5]). A sequence of random variables is said to be linearly negative quadrant dependent (LNQD, in short) if, for any disjoint subsets and positive ,

Remark 1.5. It is easily seen that if is a sequence of LNQD random variables, then is still a sequence of LNQD random variables, where and are real numbers.

The NA property has aroused wide interest because of numerous applications in reliability theory, percolation theory, and multivariate statistical analysis. In the past decades, a lot of effort was dedicated to proving the limit theorems of NA random variables. A Kolmogorov-type strong law of large numbers of NA random variables was established by Matuła in [6], which is the same as I.I.D. sequence, and Marcinkiewicz-type strong law of large Numbers was obtained by Su and Wang [7] for NA random variable sequence with assumptions of identical distribution; Yang et al. [8] gave the strong law of large Numbers of a general method.

The concept of LNQD sequence was introduced by Newman [5]. Some applications for LNQD sequence have been found. See, for example, Newman [5] who established the central limit theorem for a strictly stationary LNQD process. Wang and Zhang [9] provided uniform rates of convergence in the central limit theorem for LNQD sequence. Ko et al. [10] obtained the Hoeffding-type inequality for LNQD sequence. Ko et al. [11] studied the strong convergence for weighted sums of LNQD arrays, and so forth.

The aim of this paper is to establish a strong law of large numbers for arrays of NA and LNQD random variables. The result obtained not only extends Theorem 1.1 for independent sequence above to the case of NA and LNQD random variables sequence, but also improves it.

Lemma 1.6 (cf. [12]). Let be NA random variables, , , , . Then, there exists a positive constant such that

Let denote a positive constant which is not necessary the same in its each appearance.

Lemma 1.7 (cf. [3, 4]). Let random variables and be NQD, then

Lemma 1.8. Let be LNQD random variables sequences with mean zero and . Then, for any , .

This lemma is easily proved by following Fuk and Nagaev [13]. Here, we omit the details of the proof.

2. Main Results

Theorem 2.1. Let be an array of rowwise NA random variables. Let be a sequence of positive real numbers such that . Let be a positive, even function such that is an increasing function of and is a decreasing function of , respectively, that is, for some nonnegative integer . If and where is a positive integer and , then

Proof of Theorem 2.1. For all , , , then, for all , First, we show that In fact, by , as and , then From (2.4) and (2.5), it follows that, for sufficiently large, Hence, we need only to prove that From the fact that , it follows easily that By and as , then as .
By the Markov inequality, Lemma 1.6, and , we have Now we complete the proof of Theorem 2.1.

Corollary 2.2. Under the conditions of Theorem 2.1, then

Proof of Corollary 2.2. By Theorem 2.1, the proof of Corollary 2.2 is obvious.

Remark 2.3. Corollary 2.2 not only generalizes the result of Hu and Taylor to NA random variables, but also improves it.

Theorem 2.4. Let be an array of rowwise LNQD random variables. Let be a sequence of positive real numbers such that . Let be a positive, even function such that is an increasing function of and is a decreasing function of , respectively, that is, for some positive integer . If and then

Proof of Theorem 2.4. For any , , let To prove (2.14), it suffices to show that
Firstly, we prove (2.16):
Secondly, we prove (2.17). By Lemma 1.7, we know that is an array of rowwise LNQD mean zero random variables. Let . Take , , and . By Lemma 1.8, for all , From (2.12), (2.13), the Markov inequality, and -inequality, Note that , and . From (2.12), (2.13), and the -inequality,
Finally, we prove (2.18). For , , , then . From the definition of if , then , if , then . So . Consequently, The proof is completed.

Theorem 2.5. Let be an array of rowwise LNQD random variables. Let , be a sequence of positive real numbers such that . Let , be a sequence of positive even functions and satisfy (2.12) for . Suppose that where is a positive integer, , the conditions (2.13) and (2.24) imply (2.24).

Proof of Theorem 2.5. Following the notations and the methods of the proof in Theorem 2.4, (2.16), (2.18), and hold. So, we only need to show that . Let . By (2.24), we have The proof is completed.

Corollary 2.6. Under the conditions of Theorem 2.4 or Theorem 2.5, then

Remark 2.7. Because of the maximal inequality of LNQD, the result of LNQD we have obtained generalizes and improves the result of Hu and Taylor.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (11061012), the Guangxi China Science Foundation (2011GXNSFA018147), and the Innovation Project of Guangxi Graduate Education (2010105960202M32).

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