- Research Article
- Open Access

# Moment Inequality for -Mixing Sequences and Its Applications

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

**2009**:379743

https://doi.org/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.

## Keywords

- Growth Rate
- Positive Integer
- Variable Sequence
- Probability Space
- Simple Fact

## 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]).

for every and .

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

Theorem 2.1.

where may be different in various places.

Proof.

The proof is completed.

## 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.

where is defined in (3.1).

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

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## Copyright

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