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# Moment Inequality for -Mixing Sequences and Its Applications

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 379743 (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

Let be a random variable sequence defined on a fixed probability space and for each . Let and be positive integers. Write . Given -algebras , in , let

Define the -mixing coefficients by

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]).

Let be a sequence of -mixing random variables. Let , , , and . Then

Lemma 1.3 (see Shao [10]).

Let be a -mixing sequence. Put . Suppose that there exists an array of positive numbers such that

Then for every , there exists a constant depending only on and such that

for every and .

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

Let be a nondecreasing unbounded sequence of positive numbers and let be nonnegative numbers. Let and be fixed positive numbers. Assume that for each ,

then

and with the growth rate

where

If further we assume that for infinitely many , then

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

Let be a nondecreasing unbounded sequence of positive numbers and let be nonnegative numbers. Denote for . Let be a fixed positive number satisfying (1.6). If

then (1.8)–(1.11) hold.

## 2. Maximal Inequality for -Mixing Sequences

Theorem 2.1.

Let be a sequence of -mixing random variables satisfying . Assume that and for each . Then there exists a constant depending only on such that for any and

In particular,

where may be different in various places.

Proof.

By Lemma 1.2 for , we can see that

which implies (2.1). By (2.1) and Lemma 1.3 (take ), we can get

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.

Let be a sequence of -mixing random variables satisfying and let be a nondecreasing sequence of positive numbers. Then for any and any integer ,

where is defined in (2.4) in Theorem 2.1.

Proof.

Without loss of generality, we assume that for all . Let . For , define

For , we let and be the index of the last nonempty set . Obviously, if and . It is easily seen that if and is also a sequence of -mixing random variables. By Markov's inequality and (2.4), we have

Now we estimate . Let . Then follows from the definition of . Therefore,

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

Theorem 3.2.

Let be a sequence of -mixing random variables satisfying and let be a nondecreasing sequence of positive numbers. Then for any and any positive integers ,

where is defined in (3.1).

Proof.

Observe that

thus

For , by Markov's inequality and (2.3), we have

For , we will apply Theorem 3.1 to and . Noting that

thus, by Theorem 3.1, we get

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

Theorem 3.3.

Let be a sequence of -mixing random variables satisfying and let be a nondecreasing sequence of positive numbers. Denote for . Assume that

then for any ,

where is defined in (3.1). Furthermore, if , then

Proof.

By the continuity of probability and Theorem 3.1, we get

Observe that

By Theorem 3.2, we have that

Hence, by (3.11) and Kronecker's Lemma, it follows that

which is equivalent to

So the desired results are proved.

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

Theorem 4.1.

Let be a sequence of mean zero -mixing random variables satisfying and let be a nondecreasing unbounded sequence of positive numbers. Assume that

then

and with the growth rate

where

If further we assume that for infinitely many , then

Proof.

By (2.4) in Theorem 2.1, we have

It follows by (4.1) that

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.

Let be a sequence of -mixing random variables with . . Denote for and . Assume that

then

and with the growth rate

where

If further we assume that for infinitely many , then

In addition, for any ,

Proof.

Assume that , and , . By (2.4) in Theorem 2.1, we can see that

It is a simple fact that for all . It follows by (4.9) that

That is to say (1.12) holds. By Remark in Fazekas and Klesov [14], (1.12) implies (1.13). By Lemma 1.5, we can obtain (4.10)–(4.15) immetiately. By (4.14), it follows that

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).

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## 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.

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Xuejun, W., Shuhe, H., Yan, S. *et al.* Moment Inequality for -Mixing Sequences and Its Applications.
*J Inequal Appl* **2009**, 379743 (2009). https://doi.org/10.1155/2009/379743

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DOI: https://doi.org/10.1155/2009/379743

### Keywords

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