- Xing-Cai Zhou
^{1, 2}, - Chang-Chun Tan
^{3}Email author and - Jin-Guan Lin
^{1}

**2011**:157816

https://doi.org/10.1155/2011/157816

© Xing-Cai Zhou et al. 2011

**Received: **26 October 2010

**Accepted: **27 January 2011

**Published: **22 February 2011

## Abstract

## 1. Introduction

Definition 1.1.

A random variable sequence is said to be a -mixing random variable sequence if there exists such that .

The notion of -mixing seems to be similar to the notion of -mixing, but they are quite different from each other. Many useful results have been obtained for -mixing random variables. For example, Bradley [1] has established the central limit theorem, Byrc and Smoleński [2] and Yang [3] have obtained moment inequalities and the strong law of large numbers, Wu [4, 5], Peligrad and Gut [6], and Gan [7] have studied almost sure convergence, Utev and Peligrad [8] have established maximal inequalities and the invariance principle, An and Yuan [9] have considered the complete convergence and Marcinkiewicz-Zygmund-type strong law of large numbers, and Budsaba et al. [10] have proved the rate of convergence and strong law of large numbers for partial sums of moving average processes based on -mixing random variables under some moment conditions.

For a sequence of . random variables, Baum and Katz [11] proved the following well-known complete convergence theorem: suppose that is a sequence of . random variables. Then and if and only if for all .

Hsu and Robbins [12] and Erdös [13] proved the case and of the above theorem. The case and of the above theorem was proved by Spitzer [14]. An and Yuan [9] studied the weighted sums of identically distributed -mixing sequence and have the following results.

Theorem B.

Theorem C.

Recently, Sung [15] obtained the following complete convergence results for weighted sums of identically distributed NA random variables.

Theorem D.

We find that the proof of Theorem C is mistakenly based on the fact that (1.5) holds for . Hence, the Marcinkiewicz-Zygmund-type strong laws for -mixing sequence have not been established.

In this paper, we shall not only partially generalize Theorem D to -mixing case, but also extend Theorem B to the case . The main purpose is to establish the Marcinkiewicz-Zygmund strong laws for linear statistics of -mixing random variables under some suitable conditions.

We have the following results.

Theorem 1.2.

where for some and . Let . If for and (1.8) for , then (1.9) holds.

Remark 1.3.

The proof of Theorem D was based on Theorem 1 of Chen et al. [16], which gave sufficient conditions about complete convergence for NA random variables. So far, it is not known whether the result of Chen et al. [16] holds for -mixing sequence. Hence, we use different methods from those of Sung [15]. We only extend the case of Theorem D to -mixing random variables. It is still open question whether the result of Theorem D about the case holds for -mixing sequence.

Theorem 1.4.

## 2. Proof of the Main Result

Throughout this paper, the symbol represents a positive constant though its value may change from one appearance to next. It proves convenient to define , where denotes the natural logarithm.

To obtain our results, the following lemmas are needed.

Lemma 2.1 (Utev and Peligrad [8]).

Suppose is a positive integer, , and . Then there exists a positive constant such that the following statement holds.

Lemma 2.2.

Proof.

Hence, (1.7) is satisfied. From the proof of (2.1) of Sung [15], we obtain easily that the result holds.

Proof of Theorem 1.2.

To obtain (1.9), we need only to prove that and .

For , we consider the following two cases.

Next, we prove in the following two cases.

If or , take . Noting that , we have

If or , one gets . Since , it implies . Therefore, we have

Proof of Theorem 1.4.

## Declarations

### Acknowledgments

The authors thank the Academic Editor and the reviewers for comments that greatly improved the paper. This work is partially supported by Anhui Provincial Natural Science Foundation (no. 11040606M04), Major Programs Foundation of Ministry of Education of China (no. 309017), National Important Special Project on Science and Technology (2008ZX10005-013), and National Natural Science Foundation of China (11001052, 10971097, and 10871001).

## Authors’ Affiliations

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