Open Access

Moment Inequality for -Mixing Sequences and Its Applications

Journal of Inequalities and Applications20092009:379743

DOI: 10.1155/2009/379743

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

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
(1.1)
Define the -mixing coefficients by
(1.2)

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
(1.3)

Lemma 1.3 (see Shao [10]).

Let be a -mixing sequence. Put . Suppose that there exists an array of positive numbers such that
(1.4)
Then for every , there exists a constant depending only on and such that
(1.5)

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 ,
(1.6)
(1.7)
then
(1.8)
and with the growth rate
(1.9)
where
(1.10)
If further we assume that for infinitely many , then
(1.11)

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
(1.12)
(1.13)

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
(2.1)
(2.2)
In particular,
(2.3)
(2.4)

where may be different in various places.

Proof.

By Lemma 1.2 for , we can see that
(2.5)
which implies (2.1). By (2.1) and Lemma 1.3 (take ), we can get
(2.6)

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 ,
(3.1)

where is defined in (2.4) in Theorem 2.1.

Proof.

Without loss of generality, we assume that for all . Let . For , define
(3.2)
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
(3.3)
Now we estimate . Let . Then follows from the definition of . Therefore,
(3.4)

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 ,
(3.5)

where is defined in (3.1).

Proof.

Observe that
(3.6)
thus
(3.7)
For , by Markov's inequality and (2.3), we have
(3.8)
For , we will apply Theorem 3.1 to and . Noting that
(3.9)
thus, by Theorem 3.1, we get
(3.10)

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
(3.11)
then for any ,
(3.12)
where is defined in (3.1). Furthermore, if , then
(3.13)

Proof.

By the continuity of probability and Theorem 3.1, we get
(3.14)
Observe that
(3.15)
By Theorem 3.2, we have that
(3.16)
Hence, by (3.11) and Kronecker's Lemma, it follows that
(3.17)
which is equivalent to
(3.18)

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
(4.1)
then
(4.2)
and with the growth rate
(4.3)
where
(4.4)
(4.5)
If further we assume that for infinitely many , then
(4.6)

Proof.

By (2.4) in Theorem 2.1, we have
(4.7)
It follows by (4.1) that
(4.8)

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
(4.9)
then
(4.10)
and with the growth rate
(4.11)
where
(4.12)
(4.13)
(4.14)
If further we assume that for infinitely many , then
(4.15)
In addition, for any ,
(4.16)

Proof.

Assume that , and , . By (2.4) in Theorem 2.1, we can see that
(4.17)
It is a simple fact that for all . It follows by (4.9) that
(4.18)
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
(4.19)

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

(1)
School of Mathematical Science, Anhui University

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© Wang Xuejun et al. 2009

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.