- Research Article
- Open Access

- Wang Xuejun
^{1}, - Hu Shuhe
^{1}Email author, - Yang Wenzhi
^{1}and - Shen Yan
^{1}

**2010**:372390

https://doi.org/10.1155/2010/372390

© Wang Xuejun et al. 2010

**Received:**4 February 2010**Accepted:**11 June 2010**Published:**6 July 2010

## Abstract

## Keywords

- Positive Integer
- Real Number
- Limit Theorem
- Probability Space
- Central Limit

## 1. Introduction

Definition 1.1.

A random variable sequence
*,*
is said to be a
-mixing random variable sequence if
as
*.*

-mixing random variables were introduced by Dobrushin [1] and many applications have been found. See, for example, Dobrushin [1], Utev [2], and Chen [3] for central limit theorem, Herrndorf [4] and Peligrad [5] for weak invariance principle, Sen [6, 7] for weak convergence of empirical processes, Shao [8] for almost sure invariance principles, Hu and Wang [9] for large deviations, and so forth. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.

Throughout the paper, let be the indicator function of the set . We assume that is a positive increasing function on satisfying as and is the inverse function of . Since , it follows that . For easy notation, we let and . denotes that there exists a positive constant such that . denotes a positive constant which may be different in various places.

Let , be a sequence of random variables and let , be an array of constants. The almost sure limiting behavior of weighted sums was studied by many authors; see, for example, Choi and Sung [10], Cuzick [11], Wu [12], and Sung [13, 14], and so forth.

The main purpose of this paper is to extend the complete convergence for weighted sums of random variables to the case of -mixing random variables.

Definition 1.2.

Definition 1.3.

for all , where is a positive constant.

Lemma 1.4.

Lemma 1.5 (cf. [15, Lemma ]).

Lemma 1.6 (cf. [8, Lemma 2.2]).

Lemma 1.7.

Proof.

which implies (1.7). By Lemma 1.6, we can get the desired result (1.9) immediately. The proof is complete.

Lemma 1.8.

Proof.

The proof is similar to that of Lemma by Sung [14]. So we omit it.

## 2. Main Results and Their Proofs

Theorem 2.1.

Let , be a sequence of identically distributed -mixing random variables satisfying , , , and . Assume that the inverse function of satisfies (1.12). Let , , be an array of constants such that

Proof.

We complete the proof of the theorem.

Theorem 2.2.

Proof.

Note that if the series is convergent, then (2.15) holds. Therefore, we will consider only such sequences , for which the series is divergent.

Therefore, we can conclude that (2.15) holds by (2.12), (2.13), (2.14), and (2.19).

Theorem 2.3.

Proof.

following from and . We get the desired result by Theorem 2.2 immediately. The proof is completed.

Theorem 2.4.

for some and . Then for any and , (2.21) holds.

Proof.

Theorem 2.5.

Proof.

In order to prove that (2.14) holds, we should consider the following two cases.

Thus we get the desired result.

## Declarations

### Acknowledgments

The authors are most grateful to the Editor Andrei I. Volodin and anonymous referee for the careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (10871001, 60803059), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Talents Youth Fund of Anhui Province Universities (2010SQRL016ZD), and Youth Science Research Fund of Anhui University (2009QN011A).

## Authors’ Affiliations

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