- Research Article
- Open Access

# On Strong Law of Large Numbers for Dependent Random Variables

- Zhongzhi Wang
^{1}Email author

**2011**:279754

https://doi.org/10.1155/2011/279754

© Zhongzhi Wang. 2011

**Received:**16 December 2010**Accepted:**3 March 2011**Published:**14 March 2011

## Abstract

We discuss strong law of large numbers and complete convergence for sums of uniformly bounded negatively associate (NA) random variables (RVs). We extend and generalize some recent results. As corollaries, we investigate limit behavior of some other dependent random sequence.

## Keywords

- Probability Space
- Nonnegative Integer
- Moment Condition
- Generic Constant
- Disjoint Subset

## 1. Introduction

Throughout this paper, let denote the set of nonnegative integer, let be a sequence of random variables defined on probability space ( ), and put . The symbol will denote a generic constant ( ) which is not necessarily the same one in each appearance.

The main result of Jajte is as follows.

Theorem 1.1.

Let be a positive, increasing function and a positive function such that satisfies the following conditions.

(1)For some , is strictly increasing on with range .

(2)There exist C and a positive integer such that , .

(3)There exist constants and such that , .

where is the inverse of function , is the indicator of event .

Motivated by Jajte [1], the present paper is devoted to the study of the limiting behavior of sums when are dependent RVs In particular, we willl consider the case when are NA RVs and obtain some general results on the complete convergence of dependent RVs First, we shall give some definitions.

Definition 1.2.

whenever and are coordinatewise increasing and the covariance exists.

Definition 1.3.

whenever and are coordinatewise increasing and the covariance exists.

An infinite family of random variables is NA (resp., PA) if every finite subfamily is NA (resp., PA).

Let be the -algebra generated by RVs , .

Definition 1.4.

A sequence of random variables is said to be -dependence if and are independent for all and such that .

Definition 1.5.

These concepts of dependence were introduced by Esary et al. [2] and Joag-Dev and Proschan [3]. Their basic properties may be found in [2, 3] and the references therein.

Definition 1.6.

Remark 1.7.

then there exists a uniformly bounded random variable such that .

The structure of this paper is as follows. Some needed technical results will be presented in Section 2. The strong law of large numbers for NA RVs will be established in Section 3. The Spitzer and Hus-Robbins-type law of large numbers will be presented in Sections 4 and 5, respectively.

## 2. Preliminaries

We now present some terminologies and lemmas. The following six properties are listed for reference in obtaining the main results in the next sections. Detailed proofs can be founded in the cited references.

Lemma 2.1 (cf. [4]) (three-series theorem for NA).

Let be NA. Let and let . In order that converges a.s., it is sufficient that

(1) ,

(2) converges,

(3) .

Lemma 2.2 (cf. [4]).

Lemma 2.3 (cf. [5]).

Lemma 2.4 (cf. [5]).

Lemma 2.5 (cf. [6]).

where .

Lemma 2.6.

Proof.

Lemma 2.7 (cf. [7]).

Let be a sequence of events defined on . If , then , if and for , then .

## 3. Strong Law of Large Numbers

Theorem 3.1.

Conversely, let be a sequence of identically distributed NA random variables, if (3.1) is true, then .

Proof.

Here, we shall use the three-series theorem for NA RVs.

where and .

which is equivalent to .

These complete the proof of Theorem 3.1.

Theorem 3.1 also includes a particular case of logarithmic means, we can establish the following.

Corollary 3.2.

Proof.

Let , , that is, . In this case, as , therefore , for .

Corollary 3.3.

Remark 3.4.

As pointed out by Jajte [1], Theorem 3.1 includes several regular summability methods such as (1) the Kolmogorov SLLN ; (2) the classical MZ SLLN .

## 4. Spitzer Type Law of Large Numbers

Since the definition of complete convergence was introduced by Hsu and Robins, there have been many authors who devote themselves to the study of the complete convergence for sums of independent and dependent RVs and obtain a series of elegant results, see [4, 8] and reference therein.

Theorem 4.1.

Conversely, let be a sequence of identically distributed NA random variables, if (4.3) is true, then and .

Proof.

Therefore, (4.3) follows.

Conversely, since imply that ., hence from Theorem 3.1, we have . These complete the proof of Theorem 4.1.

Corollary 4.2.

Proof.

hence, similarly to the proof of Theorem 4.1, we obtain (4.10).

Analogously, we can prove the following corollaries, and omit the details.

Corollary 4.3.

then (4.3) holds.

Corollary 4.4.

Let be a sequence of -dependent random variables with . Assume that . If , then (4.3) holds.

Corollary 4.5.

then (4.3) holds.

## 5. Hsu-Robbins Type Law of Large Numbers

Theorem 5.1.

Let , be define as in Theorem 1.1, but the following condition (3) is replaced by

(3′) There exist constants such that , , , and let be a sequence of NA random variables with . Assume that . If , then for all ,

Proof.

From the previous section, we know that to prove Theorem 5.1, we need only to prove the convergence of the following three series.

These complete the proof of Theorem 5.1.

## Declarations

### Acknowledgments

This work is supported by the National Nature Science Foundation of China (no. 11071104) and the Anhui high Education Research Grant (no. KJ2010A337). The author expresses his sincere gratitude to the referees and the editors for their hospitality.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.