Journal of Probability and Statistics

Volume 2011 (2011), Article ID 708087, 10 pages

http://dx.doi.org/10.1155/2011/708087

## 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

Academic Editor: Man Lai Tang

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).

#### References

- T. C. Hu and R. L. Taylor, “On the strong law for arrays and for the bootstrap mean and variance,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 20, no. 2, pp. 375–382, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Joag-Dev and F. Proschan, “Negative association of random variables, with applications,”
*Annals of Statistics*, vol. 11, no. 1, pp. 286–295, 1983. View at Publisher · View at Google Scholar - E. L. Lehmann, “Some concepts of dependence,”
*Annals of Mathematical Statistics*, vol. 37, pp. 1137–1153, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. L. Lehmann, “Some concepts of dependence,”
*Annals of Mathematical Statistics*, vol. 78, pp. 794–803, 1966. View at Google Scholar - C. M. Newman, “Asymptotic independence and limit theorems for positively and negatively dependent random variables,” in
*Inequalities in Statistics and Probability*, Y. L. Tong, Ed., vol. 5 of*IMS Lecture Notes—Monograph Series*, pp. 127–140, Institute of Mathematical Statistics, Hayward, Calif, USA, 1984. View at Publisher · View at Google Scholar - P. Matuła, “A note on the almost sure convergence of sums of negatively dependent random variables,”
*Statistics and Probability Letters*, vol. 15, no. 3, pp. 209–213, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Su and Y. B. Wang, “The strong convergence of the identical distribution NA sequence,”
*Chinese Journal of Applied Probability and Statistics*, vol. 14, no. 2, pp. 131–140, 1998. View at Google Scholar · View at Zentralblatt MATH - S. Yang, C. Su, and K. Yu, “A general method to the strong law of large numbers and its applications,”
*Statistics and Probability Letters*, vol. 78, no. 6, pp. 794–803, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. F. Wang and L. X. Zhang, “A Berry-Esseen theorem for weakly negatively dependent random variables and its applications,”
*Acta Mathematica Hungarica*, vol. 110, no. 4, pp. 293–308, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. H. Ko, Y. K. Choi, and Y. S. Choi, “Exponential probability inequality for linearly negative quadrant dependent random variables,”
*Korean Mathematical Society. Communications*, vol. 22, no. 1, pp. 137–143, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. H. Ko, D. H. Ryu, and T. S. Kim, “Limiting behaviors of weighted sums for linearly negative quadrant dependent random variables,”
*Taiwanese Journal of Mathematics*, vol. 11, no. 2, pp. 511–522, 2007. View at Google Scholar · View at Zentralblatt MATH - 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 - D. H. Fuk and S. V. Nagaev, “Probabilistic inequalities for sums of independent random variables,”
*Theory of Probability and Its Applications*, vol. 16, pp. 643–660, 1971. View at Google Scholar