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Journal of Probability and Statistics
Volume 2011 (2011), Article ID 708087, 10 pages
http://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

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 {𝑋𝑛}𝑛 be a sequence of independent distributed random variables. The Marcinkiewicz-Zygmund strong law of large numbers (SLLN) provides that1𝑛𝑛1/𝛼𝑖=1𝑋𝑖𝐸𝑋𝑖10a.s.for1𝛼<2,𝑛𝑛1/𝛼𝑖=1𝑋𝑖0a.s.for0<𝛼<1as𝑛(1.1) if and only if 𝐸|𝑋|𝛼<. The case 𝛼=1 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 {𝑋𝑛𝑖,1𝑖𝑛,𝑛1} be a triangular array of rowwise independent random variables. Let {𝑎𝑛}𝑛 be a sequence of positive real numbers such that 0<𝑎𝑛. Let 𝜓(𝑡) be a positive, even function such that 𝜓(𝑡)/|𝑡|𝑝 is an increasing function of |𝑡| and 𝜓(𝑡)/|𝑡|𝑝+1 is a decreasing function of |𝑡|, respectively, that is, 𝜓(𝑡)|𝑡|𝑝,𝜓(𝑡)|𝑡|𝑝+1,as|𝑡|,(1.2) for some positive integer 𝑝. If 𝑝2 and 𝐸𝑋𝑛𝑖=0,𝑛𝑛=1𝑖=1𝐸𝜓||𝑋𝑛𝑖||𝜓||𝑎𝑛||<,𝑛=1𝑛𝑖=1𝐸𝑋𝑛𝑖𝑎𝑛22𝑘<,(1.3) where 𝑘 is a positive integer, then 1𝑎𝑛𝑛𝑖=1𝑋𝑛𝑖0a.s.(1.4)

Definition 1.2 (cf. [2]). A finite family of random variables {𝑋𝑛}𝑛 is said to be negatively associated (NA, in short) if, for any disjoint subsets 𝐴 and 𝐵 of {1,2,,𝑛} and any real coordinate-wise nondecreasing functions 𝑓 on on 𝐴 and 𝑔 on 𝐵, 𝑓𝑋Cov𝑖𝑌,𝑖𝐴,𝑔𝑗,𝑗𝐵0,(1.5) 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 𝑋 and 𝑌 are said to be negative quadrant dependent (NQD, in short) if, for any 𝑥,𝑦, 𝑃(𝑋<𝑥,𝑌<𝑦)𝑃(𝑋<𝑥)𝑃(𝑌<𝑦).(1.6) A sequence {𝑋𝑛}𝑛 of random variables is said to be pairwise NQD if 𝑋𝑖 and 𝑋𝑗 are NQD for all 𝑖,𝑗+ and 𝑖𝑗.

Definition 1.4 (cf. [5]). A sequence {𝑋𝑛}𝑛 of random variables is said to be linearly negative quadrant dependent (LNQD, in short) if, for any disjoint subsets 𝐴,𝐵+ and positive 𝑟𝑗𝑠, 𝑘𝐴𝑟𝑘𝑋𝑘,𝑗𝐵𝑟𝑗𝑋𝑗areNQD.(1.7)

Remark 1.5. It is easily seen that if {𝑋𝑛}𝑛 is a sequence of LNQD random variables, then {𝑎𝑋𝑛+𝑏}𝑛 is still a sequence of LNQD random variables, where 𝑎 and 𝑏 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 {𝑋𝑛,𝑛1} be NA random variables, 𝐸𝑋𝑛=0, 𝐸|𝑋𝑛|𝑞<, 𝑛1, 𝑞2. Then, there exists a positive constant 𝑐 such that 𝐸max1𝑖𝑛|||||𝑘𝑖=1𝑋𝑖|||||𝑞𝑐𝑛𝑖=1𝐸||𝑋𝑖||q+𝑛𝑖=1𝐸𝑋𝑖2𝑞/2,𝑛1.(1.8)

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

Lemma 1.7 (cf. [3, 4]). Let random variables 𝑋 and 𝑌 be NQD, then (1)𝐸𝑋𝑌𝐸𝑋𝐸𝑌;(2)𝑃(𝑋<𝑥,𝑌<𝑦)𝑃(𝑋<𝑥)𝑃(𝑌<𝑦);(3)if𝑓and𝑔arebothnondecreasing(orbothnonincreasing)functions,then𝑓(𝑋)and𝑔(𝑌)areNQD.(1.9)

Lemma 1.8. Let {𝑋𝑛,𝑛1} be LNQD random variables sequences with mean zero and 0<𝐵𝑛=𝑛𝑘=1𝐸𝑋2𝑘. Then, 𝑃||𝑆𝑛||𝑥𝑛𝑘=1𝑃||𝑆𝑘||𝑥𝑦+2exp𝑦𝑥𝑦log1+𝑥𝑦𝐵𝑛,(1.10) for any 𝑥>0, 𝑦>0.

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 {𝑋𝑛𝑖;𝑖1,𝑛1} be an array of rowwise NA random variables. Let {𝑎𝑛}𝑛 be a sequence of positive real numbers such that 0<𝑎𝑛. Let 𝜓(𝑡) be a positive, even function such that 𝜓(𝑡)/|𝑡| is an increasing function of |𝑡| and 𝜓(𝑡)/|𝑡|𝑝 is a decreasing function of |𝑡|, respectively, that is, 𝜓(𝑡)|𝑡|,𝜓(𝑡)|𝑡|𝑝,as|𝑡|(2.1) for some nonnegative integer 𝑃. If 𝑝2 and 𝐸𝑋𝑛𝑖=0,𝑛𝑛=1𝑖=1𝐸𝜓||𝑋𝑛𝑖||𝜓||𝑎𝑛||<,𝑛=1𝑛𝑖=1𝐸𝑋𝑛𝑖𝑎𝑛2𝑣/2<,(2.2) where 𝑣 is a positive integer and 𝑣𝑝, then 𝑛=1𝑃max1𝑘𝑛|||||1𝑎𝑛𝑘𝑖=1𝑋𝑛𝑖|||||>𝜀<,forany𝜀>0.(2.3)

Proof of Theorem 2.1. For all 𝑖1, let𝑋𝑖(𝑛)=𝑎𝑛𝐼(𝑋𝑛𝑖<𝑎𝑛)+𝑋𝑛𝑖𝐼(|𝑋𝑛𝑖|𝑎𝑛)+𝑎𝑛𝐼(𝑋𝑛𝑖>𝑎𝑛), 𝑇𝑗(𝑛)=(1/𝑎𝑛)𝑗𝑖=1(𝑋𝑖(𝑛)𝐸𝑋𝑖(𝑛)), then, for all 𝜀>0, 𝑃max1𝑘𝑛|||||1𝑎𝑛𝑘𝑖=1𝑋𝑛𝑖|||||>𝜀𝑃max1𝑗𝑛||𝑋𝑛𝑗||>𝑎𝑛+𝑃max1𝑗𝑛||𝑇𝑗(𝑛)||>𝜀max1𝑗𝑛|||||1𝑎𝑛𝑗𝑖=1𝐸𝑋𝑖(𝑛)|||||.(2.4) First, we show that max1𝑗𝑛|||||1𝑎𝑛𝑗𝑖=1𝐸𝑋𝑖(𝑛)|||||0,as𝑛.(2.5) In fact, by 𝐸𝑋𝑛𝑖=0, 𝜓(𝑡)/|𝑡| as |𝑡| and Σ𝑛=1Σ𝑛𝑖=1𝐸(𝜓(|𝑋𝑛𝑖|)/𝜓(𝑎𝑛))<, then max1𝑗𝑛|||||1𝑎𝑛𝑗𝑖=1𝐸𝑋𝑖(𝑛)|||||max1𝑗𝑛1𝑎𝑛|||||𝑗𝑖=1𝐸𝑋𝑛𝑖𝐼||𝑋𝑛𝑖||𝑎𝑛|||||+|||||𝑗𝑖=1𝐸𝑎𝑛𝐼||𝑋𝑛𝑖||>𝑎𝑛|||||max1𝑗𝑛1𝑎𝑛𝑗𝑖=1||𝐸𝑋𝑛𝑖𝐼||𝑋𝑛𝑖||𝑎𝑛||+|||||𝑗𝑖=1𝐸𝑎𝑛𝐼||𝑋𝑛𝑖||>𝑎𝑛|||||=max1𝑗𝑛1𝑎𝑛𝑗𝑖=1||𝐸𝑋𝑛𝑖𝐼||𝑋𝑛𝑖||>𝑎𝑛||+|||||𝑗𝑖=1𝐸𝑎𝑛𝐼||𝑋𝑛𝑖||>𝑎𝑛|||||2𝑛𝑖=1𝐸||𝑋𝑛𝑖||𝐼||𝑋𝑛𝑖||>𝑎𝑛𝑎𝑛2𝑛𝑖=1||𝑋𝐸𝜓𝑛𝑖||𝐼||𝑋𝑛𝑖||>𝑎𝑛𝜓𝑎𝑛2𝑛𝑖=1||𝑋𝐸𝜓𝑛𝑖||𝜓𝑎𝑛0,as𝑛.(2.6) From (2.4) and (2.5), it follows that, for 𝑛 sufficiently large, 𝑃max1𝑘𝑛|||||1𝑎𝑛𝑘𝑖=1𝑋𝑛𝑖|||||>𝜀𝑛𝑗=1𝑃||𝑋𝑛𝑗||>𝑎𝑛+𝑃max1𝑗𝑛||𝑇𝑗(𝑛)||>𝜀2.(2.7) Hence, we need only to prove that 𝐼=𝑛𝑛=1𝑗=1𝑃||𝑋𝑛𝑗||>𝑎𝑛<,𝐼𝐼=𝑛=1𝑃max1𝑗𝑛||𝑇𝑗(𝑛)||>𝜀2<.(2.8) From the fact that Σ𝑛=1Σ𝑛𝑖=1𝐸(𝜓(|𝑋𝑛𝑖|)/𝜓(𝑎𝑛))<, it follows easily that 𝐼=𝑛𝑛=1𝑗=1𝑃||𝑋𝑛𝑗||>𝑎𝑛𝑛𝑛=1𝑗=1𝐸𝜓||𝑋𝑛𝑗||𝜓𝑎𝑛<.(2.9) By 𝑣𝑝 and 𝜓(𝑡)/|𝑡|𝑝 as |𝑡|, then 𝜓(𝑡)/|𝑡|𝑣 as |𝑡|.
By the Markov inequality, Lemma 1.6, and 𝑛=1(𝑛𝑖=1𝐸(𝑋𝑛𝑖/𝑎𝑛)2)𝑣/2<, we have 𝐼𝐼=𝑛=1𝑃max1𝑗𝑛||𝑇𝑗(𝑛)||>𝜀2𝑛=1𝜀2𝑣𝐸max1𝑗𝑛||𝑇𝑗(𝑛)||𝑣𝜀𝑐2𝑣𝑛=11𝑎𝑣𝑛𝑛𝑗=1𝐸||𝑋𝑗(𝑛)||2𝑣/2+𝑛𝑗=1𝐸||𝑋𝑗(𝑛)||𝑣𝑐𝑛=11𝑎𝑣𝑛𝑛𝑗=1𝐸||𝑋𝑗(𝑛)||𝑣+𝑐𝑛=11𝑎𝑣𝑛𝑛𝑗=1𝐸||𝑋𝑗(𝑛)||2𝑣/2𝑐𝑛=11𝑎𝑣𝑛𝑛𝑗=1𝐸||𝑋𝑛𝑗||𝑣𝐼||𝑋𝑛𝑗||𝑎𝑛+𝐼+𝑐𝑛=11𝑎𝑣𝑛𝑛𝑗=1𝐸||𝑋𝑗(𝑛)||2𝑣/2𝑐𝑛𝑛=1𝑖=1𝐸𝜓||𝑋𝑛𝑖||𝜓𝑎𝑛+𝑐𝑛=11𝑎𝑣𝑛𝑛𝑗=1𝐸||𝑋𝑗(𝑛)||2𝑣/2𝑐𝑛𝑛=1𝑖=1𝐸𝜓||𝑋𝑛𝑖||𝜓𝑎𝑛+𝑐𝑛=1𝑛𝑗=1𝐸𝑋𝑛𝑖𝑎𝑛2𝑣/2<.(2.10) Now we complete the proof of Theorem 2.1.

Corollary 2.2. Under the conditions of Theorem 2.1, then 1𝑎𝑛𝑛𝑖=1𝑋𝑛𝑖0a.s.(2.11)

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 {𝑋𝑛𝑖;𝑖1,𝑛1} be an array of rowwise LNQD random variables. Let {𝑎𝑛}𝑛 be a sequence of positive real numbers such that 0<𝑎𝑛. Let 𝜓(𝑡) be a positive, even function such that 𝜓(𝑡)/|𝑡| is an increasing function of |𝑡| and 𝜓(𝑡)/|𝑡|𝑝 is a decreasing function of |𝑡|, respectively, that is, 𝜓(𝑡)|𝑡|,𝜓(𝑡)|𝑡|𝑝,as|𝑡|(2.12) for some positive integer 𝑝. If 1<𝑝2 and 𝐸𝑋𝑛𝑖=0,𝑛𝑛=1𝑖=1𝐸𝜓||𝑋𝑛𝑖||𝜓𝑎𝑛<,(2.13) then 𝑛=1𝑃|||||1𝑎𝑛𝑛𝑖=1𝑋𝑛𝑖|||||>𝜀<,forany𝜀>0.(2.14)

Proof of Theorem 2.4. For any 1𝑘𝑛, 𝑛1, let 𝑌𝑛𝑘=𝑎𝑛𝐼𝑋𝑛𝑘<𝑎𝑛+𝑋𝑛𝑘𝐼||𝑋𝑛𝑘||𝑎𝑛+𝑎𝑛𝐼𝑋𝑛𝑘>𝑎𝑛,𝑍𝑛𝑘=𝑋𝑛𝑘𝑌𝑛𝑘=𝑋𝑛𝑘+𝑎𝑛𝐼𝑋𝑛𝑘<𝑎𝑛+𝑋𝑛𝑘𝑎𝑛𝐼𝑋𝑛𝑘>𝑎𝑛.(2.15) To prove (2.14), it suffices to show that 1𝑎𝑛𝑛𝑘=1𝑍𝑛𝑘10completely,(2.16)𝑎𝑛𝑛𝑘=1𝑌𝑛𝑘𝐸𝑌𝑛𝑘10completely,(2.17)𝑎𝑛𝑛𝑘=1𝐸𝑌𝑛𝑘0as𝑛.(2.18)
Firstly, we prove (2.16): 𝑛=1𝑃1𝑎𝑛|||||𝑛𝑘=1𝑍𝑛𝑘|||||>𝜀𝑛=1𝐸||𝑛𝑘=1𝑍𝑛𝑘||𝑎𝑛𝜀𝐶𝑛𝑛=1𝑘=1𝐸||𝑋𝑛𝑘||𝐼||𝑋𝑛𝑘||>𝑎𝑛𝑎𝑛𝐶𝑛𝑛=1𝑖=1𝐸𝜓||𝑋𝑛𝑖||𝜓𝑎𝑛<.(2.19)
Secondly, we prove (2.17). By Lemma 1.7, we know that {𝑌𝑛𝑘𝐸𝑌𝑛𝑘,1𝑘𝑛,𝑛1} is an array of rowwise LNQD mean zero random variables. Let 𝐵𝑛=𝑛𝑘=1𝐸(𝑌𝑛𝑘𝐸𝑌𝑛𝑘)2. Take 𝑥=𝜀𝑎𝑛, 𝑦=2𝜀𝑎𝑛/𝑣, and 𝑣1. By Lemma 1.8, for all 𝜀>0, 𝑛=1𝑃1𝑎𝑛|||||𝑛𝑘=1𝑌𝑛𝑘𝐸𝑌𝑛𝑘|||||>𝜀𝑛𝑛=1𝑘=1𝑃||𝑌𝑛𝑘𝐸𝑌𝑛𝑘||>𝜀𝑎𝑛𝜂+𝐶𝑛=1𝐵𝑛𝐵𝑛+𝜀2𝑎2𝑛/𝜂𝜂=𝐼1+𝐼2.(2.20) From (2.12), (2.13), the Markov inequality, and 𝐶𝑟-inequality, 𝐼1=𝑛𝑛=1𝑘=1𝑃||𝑌𝑛𝑘𝐸𝑌𝑛𝑘||>𝜀𝑎𝑛𝜂𝐶𝑛𝑛=1𝑘=1𝐸||𝑌𝑛𝑘𝐸𝑌𝑛𝑘||𝑝𝑎𝑝𝑛𝐶𝑛𝑛=1𝑘=1𝐸||𝑌𝑛𝑘||𝑝𝑎𝑝𝑛𝐶𝑛𝑛=1𝑘=1𝐸𝜓𝑘||𝑌𝑛𝑘||𝜓𝑘𝑎𝑛𝐶𝑛𝑛=1𝑘=1𝐸𝜓𝑘||𝑋𝑛𝑘||𝜓𝑘𝑎𝑛<.(2.21) Note that |𝑌𝑛𝑘||𝑋𝑛𝑘|, 𝜂1 and 1<𝑝2. From (2.12), (2.13), and the 𝐶𝑟-inequality, 𝐼2𝐶𝑛=1𝑛𝑘=1𝑎𝑛2𝐸𝑌𝑛𝑘𝐸𝑌𝑛𝑘2𝜂𝐶𝑛=1𝑛𝑘=1𝐸||𝑌𝑛𝑘||𝑝𝑎𝑝𝑛𝜂𝐶𝑛𝑛=1𝑘=1𝐸||𝑌𝑛𝑘||𝑝𝑎𝑝𝑛𝜂𝐶𝑛𝑛=1𝑘=1𝐸𝜓𝑘||𝑌𝑛𝑘||𝜓𝑘𝑎𝑛𝜂𝐶𝑛𝑛=1𝑘=1𝐸𝜓𝑘||𝑋𝑛𝑘||𝜓𝑘𝑎𝑛𝜂<.(2.22)
Finally, we prove (2.18). For 1𝑘𝑛, 𝑛1, 𝐸𝑋𝑛𝑘=0, then 𝐸𝑌𝑛𝑘=𝐸𝑍𝑛𝑘. From the definition of 𝑍𝑛𝑘 if 𝑋𝑛𝑘>𝑎𝑛, then 0<𝑍𝑛𝑘=𝑋𝑛𝑘𝑎𝑛<𝑋𝑛𝑘, if 𝑋𝑛𝑘<𝑎𝑛, then 𝑋𝑛𝑘<𝑍𝑛𝑘=𝑋𝑛𝑘+𝑎𝑛<0. So |𝑍𝑛𝑘||𝑋𝑛𝑘|𝐼(|𝑋𝑛𝑘|>𝑎𝑛). Consequently, 1𝑎𝑛|||||𝑛𝑘=1𝐸𝑌𝑛𝑘|||||=1𝑎𝑛|||||𝑛𝑘=1𝐸𝑍𝑛𝑘|||||𝑛𝑘=1𝐸||𝑍𝑛𝑘||𝑎𝑛𝑛𝑘=1𝐸||𝑋𝑛𝑘||𝐼||𝑋𝑛𝑘||>𝑎𝑛𝑎𝑛𝑛𝑘=1𝐸𝜓𝑘𝑋𝑛𝑘𝜓𝑘𝑎𝑛0as𝑛.(2.23) The proof is completed.

Theorem 2.5. Let {𝑋𝑛𝑖;𝑖1,𝑛1} be an array of rowwise LNQD random variables. Let {𝑎𝑛}𝑛, be a sequence of positive real numbers such that 0<𝑎𝑛. Let {𝜓𝑛(𝑡)}𝑛, be a sequence of positive even functions and satisfy (2.12) for 𝑝>2. Suppose that 𝑛=1𝑛𝑖=1𝐸𝑋𝑛𝑖𝑎𝑛2𝑣/2<,(2.24) where 𝑣 is a positive integer, 𝑣𝑝, 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 𝐼1< hold. So, we only need to show that 𝐼2<. Let 𝜂>𝑣/2. By (2.24), we have 𝐼2𝑛=1𝑛𝑘=1𝑎𝑛2𝐸𝑌𝑛𝑘𝐸𝑌𝑛𝑘2𝜂𝐶𝑛=1𝑛𝑘=1𝐸𝑌2𝑛𝑘𝑎2𝑛𝑣/22𝜂/𝑣𝐶𝑛=1𝑛𝑘=1𝐸𝑋2𝑛𝑘𝑎2𝑛𝑣/22𝜂/𝑣<.(2.25) The proof is completed.

Corollary 2.6. Under the conditions of Theorem 2.4 or Theorem 2.5, then 1𝑎𝑛𝑛𝑖=1𝑋𝑛𝑖0a.s.(2.26)

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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