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

**2009**:379743

**DOI: **10.1155/2009/379743

© Wang Xuejun et al. 2009

**Received: **11 April 2009

**Accepted: **21 September 2009

**Published: **11 October 2009

## Abstract

Firstly, the maximal inequality for -mixing sequences is given. By using the maximal inequality, we study the convergence properties for -mixing sequences. The Hájek-Rényi-type inequality, strong law of large numbers, strong growth rate and the integrability of supremum for -mixing sequences are obtained.

## 1. Introduction

Definition 1.1.

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

The concept of -mixing random variables was 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, Iosifescu [8] for limit theorem, Peligrad [9] for Ibragimov-Iosifescu conjecture, Shao [10] for almost sure invariance principles, Hu and Wang [11] 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. The main purpose of this paper is to study the maximal inequality for -mixing sequences, by which we can get the Hájek-Rényi-type inequality, strong law of large numbers, strong growth rate, and the integrability of supremum for -mixing sequences.

Throughout the paper, denotes a positive constant which may be different in various places. The main results of this paper depend on the following lemmas.

Lemma 1.2 (see Lu and Lin [12]).

Lemma 1.3 (see Shao [10]).

Lemma 1.4 (see Hu et al. [13]).

Lemma 1.5 (see Fazekas and Klesov [14] and Hu [15]).

then (1.8)–(1.11) hold.

## 2. Maximal Inequality for -Mixing Sequences

## 3. Hájek-Rényi-Type Inequality for -Mixing Sequences

In this section, we will give the Hájek-Rényi-type inequality for -mixing sequences, which can be applied to obtain the strong law of large numbers and the integrability of supremum.

Theorem 3.1.

where is defined in (2.4) in Theorem 2.1.

Proof.

Thus, (3.1) follows from (3.3) and (3.4) immediately.

Theorem 3.2.

Proof.

Therefore, the desired result (3.5) follows from (3.7)–(3.10) immediately.

Theorem 3.3.

Proof.

So the desired results are proved.

## 4. Strong Law of Large Numbers and Growth Rate for -Mixing Sequences

Theorem 4.1.

Proof.

Thus, (4.2)–(4.6) follow from (4.7), (4.8), and Lemma 1.4 immediately. We complete the proof of the theorem.

Theorem 4.2.

Proof.

The proof is completed.

Remark 4.3.

By using the maximal inequality, we get the Hájek-Rényi-type inequality, the strong law of large numbers and the strong growth rate for -mixing sequences. In addition, we get some new bounds of probability inequalities for -mixing sequences, such as (3.1), (3.5), (3.12), (4.5)–(4.6), and (4.13)–(4.16).

## Declarations

### Acknowledgments

The authors are most grateful to the Editor Sever Silvestru Dragomir, the referee Professor Mihaly Bencze and an anonymous referee for careful reading of the manuscript and valuable suggestions and comments which helped to significantly improve an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (Grant nos. 10871001, 60803059) and the Innovation Group Foundation of Anhui University.

## Authors’ Affiliations

## References

- Dobrushin RL:
**The central limit theorem for non-stationary Markov chain.***Theory of Probability and Its Applications*1956,**1**(4):72–88.MathSciNetMATHGoogle Scholar - Utev SA:
**On the central limit theorem for -mixing arrays of random variables.***Theory of Probability and Its Applications*1990,**35**(1):131–139. 10.1137/1135013MathSciNetView ArticleMATHGoogle Scholar - Chen DC:
**A uniform central limit theorem for nonuniform -mixing random fields.***The Annals of Probability*1991,**19**(2):636–649. 10.1214/aop/1176990445MathSciNetView ArticleMATHGoogle Scholar - Herrndorf N:
**The invariance principle for -mixing sequences.***Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete*1983,**63**(1):97–108. 10.1007/BF00534180MathSciNetView ArticleMATHGoogle Scholar - Peligrad M:
**An invariance principle for -mixing sequences.***The Annals of Probability*1985,**13**(4):1304–1313. 10.1214/aop/1176992814MathSciNetView ArticleMATHGoogle Scholar - Sen PK:
**A note on weak convergence of empirical processes for sequences of -mixing random variables.***Annals of Mathematical Statistics*1971,**42**(6):2131–2133. 10.1214/aoms/1177693079MathSciNetView ArticleMATHGoogle Scholar - Sen PK:
**Weak convergence of multidimensional empirical processes for stationary -mixing processes.***The Annals of Probability*1974,**2**(1):147–154. 10.1214/aop/1176996760View ArticleMathSciNetMATHGoogle Scholar - Iosifescu J: Limit theorem for -mixing sequences. Proceedings of the 5th Conference on Probability Theory, September 1977 1–6.Google Scholar
- Peligrad M:
**On Ibragimov-Iosifescu conjecture for -mixing sequences.***Stochastic Processes and Their Applications*1990,**35**(2):293–308. 10.1016/0304-4149(90)90008-GMathSciNetView ArticleMATHGoogle Scholar - Shao QM:
**Almost sure invariance principles for mixing sequences of random variables.***Stochastic Processes and Their Applications*1993,**48**(2):319–334. 10.1016/0304-4149(93)90051-5MathSciNetView ArticleMATHGoogle Scholar - Hu S, Wang X:
**Large deviations for some dependent sequences.***Acta Mathematica Scientia. Series B*2008,**28**(2):295–300. 10.1016/S0252-9602(08)60030-2MathSciNetView ArticleMATHGoogle Scholar - Lu CR, Lin ZY:
*Limit Theory for Mixing Dependent Sequences*. Science Press, Beijing, China; 1997.Google Scholar - Hu S, Chen G, Wang X:
**On extending the Brunk-Prokhorov strong law of large numbers for martingale differences.***Statistics & Probability Letters*2008,**78**(18):3187–3194. 10.1016/j.spl.2008.06.017MathSciNetView ArticleMATHGoogle Scholar - Fazekas I, Klesov O:
**A general approach to the strong law of large numbers.***Theory of Probability and Its Applications*2001,**45**(3):436–449. 10.1137/S0040585X97978385MathSciNetView ArticleMATHGoogle Scholar - Hu S:
**Some new results for the strong law of large numbers.***Acta Mathematica Sinica*2003,**46**(6):1123–1134.MathSciNetMATHGoogle Scholar

## 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.