Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 430180 | 9 pages | https://doi.org/10.1155/2011/430180

A Note on the Strong Law of Large Numbers for Arrays of Rowwise ρ˜-Mixing Random Variables

Academic Editor: Carlo Piccardi
Received19 Mar 2011
Accepted12 May 2011
Published21 Jul 2011

Abstract

Let be an array of rowwise -mixing random variables. Some strong law of large numbers for arrays of rowwise -mixing random variables is studied under some simple and weak conditions.

1. Introduction

Let be a sequence of independent and identically distributed random variables. The Marcinkiewicz-Zygmund strong law of large numbers states that if and only if . In the case of independence, 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 nonnegative integer . If and where is a positive integer, then

Zhu [2] generalized and improved the result of Hu and Taylor [1] for triangular arrays of rowwise independent random variables to the case of arrays of rowwise -mixing random variables as follows.

Theorem 1.2. Let be an array of rowwise -mixing 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, , then

In the following, we will give the definitions of a -mixing sequence and the array of rowwise -mixing random variables.

Let be a sequence of random variables defined on a fixed probability space . Write . For any given -algebras , in , let Define the -mixing coefficients by Obviously, and .

Definition 1.3. A sequence of random variables is said to be a -mixing sequence if, there exists such that .

An array of random variables is called rowwise -mixing if for every , is a -mixing sequence of random variables.

The -mixing random variables were introduced by Bradley [3], and many applications have been found. -mixing is similar to -mixing, but both are quite different. Many authors have studied this concept providing interesting results and applications. See, for example, Zhu [2], An and Yuan [4], Kuczmaszewska [5], Bryc and Smoleński [6], Cai [7], Gan [8], Peligrad [9, 10], Peligrad and Gut [11], Sung [12], Utev and Peligrad [13], Wu and Jiang [14], and so on. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.

The main purpose of this paper is to further study the strong law of large numbers for arrays of rowwise -mixing random variables. We will introduce some simple conditions to prove the strong law of large numbers. The techniques used in the paper are inspired by Zhu [2].

2. Main Results

Throughout the paper, let be the indicator function of the set . denotes a positive constant which may be different in various places.

The proofs of the main results of this paper are based upon the following lemma.

Lemma 2.1 (Utev and Peligrad [13, Theorem 2.1]). Let be a -mixing sequence of random variables, , for some and for every . Then, there exists a positive constant C depending only on such that

As for arrays of rowwise -mixing random variables , we assume that the constant from Lemma 2.1 is the same for each row throughout the paper. Our main results are as follows.

Theorem 2.2. Let be an array of rowwise -mixing random variables and let be a sequence of positive real numbers. Let be a sequence of positive, even functions such that is an increasing function of and is a decreasing function of for every , respectively, that is, If then, for any ,

Proof. For fixed , define It is easy to check that for any , which implies that Firstly, we will show that Actually, by conditions as and (2.3), we have that which implies (2.8). It follows from (2.7) and (2.8) that for large enough, Hence, to prove (2.4), we only need to show that The conditions as and (2.3) yield that which implies (2.11).
By Markov's inequality, Lemma 2.1 (for ), 's inequality, as and (2.3), we can get that which implies (2.12). This completes the proof of the theorem.

Corollary 2.3. Under the conditions of Theorem 2.2,

Theorem 2.4. Let be an array of rowwise -mixing random variables and let be a sequence of positive real numbers. Let be a sequence of nonnegative, even functions such that is an increasing function of for every . Assume that there exists a constant such that for . If then for any , (2.4) holds true.

Proof. We use the same notations as that in Theorem 2.2. The proof is similar to that of Theorem 2.2.
Firstly, we will show that (2.8) holds true. In fact, by the conditions for and (2.16), we have that which implies (2.8).
According to the proof of Theorem 2.2, we only need to prove that (2.11) and (2.12) hold true.
When , we have , which yields that Hence, which implies (2.11).
By Markov's inequality, Lemma 2.1 (for ), 's inequality, for and (2.16), we can get that which implies (2.12). This completes the proof of the theorem.

Corollary 2.5. Under the conditions of Theorem 2.4,

Corollary 2.6. Let be an array of rowwise -mixing random variables and let be a positive real numbers. If there exists a constant such that then (2.4) holds true.

Proof. In Theorem 2.4, we take It is easy to check that is a sequence of nonnegative, even functions such that is an increasing function of for every . And Therefore, by Theorem 2.4, we can easily get (2.4).

Acknowledgments

The authors are most grateful to the editor Carlo Piccardi and anonymous referee for careful reading of the paper and valuable suggestions which helped to improve an earlier version of this paper. This work is supported by the National Natural Science Foundation of China (10871001) and the Academic innovation team of Anhui University (KJTD001B).

References

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Copyright © 2011 Aiting Shen and Shuhe Hu. 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.


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