Open Access

On the Strong Laws for Weighted Sums of -Mixing Random Variables

Journal of Inequalities and Applications20112011:157816

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

Received: 26 October 2010

Accepted: 27 January 2011

Published: 22 February 2011

Abstract

Complete convergence is studied for linear statistics that are weighted sums of identically distributed -mixing random variables under a suitable moment condition. The results obtained generalize and complement some earlier results. A Marcinkiewicz-Zygmund-type strong law is also obtained.

1. Introduction

Suppose that is a sequence of random variables and is a subset of the natural number set . Let ,
(11)
where
(12)

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.

Let be a -mixing sequence of identically distributed random variables, , , and suppose that for . Assume that is an array of real numbers satisfying
(13)
(14)
If , then
(15)

Theorem C.

Let be a -mixing sequence of identically distributed random variables, , , and for . Assume that is array of real numbers satisfying (1.3). Then
(16)

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

Theorem D.

Let be a sequence of identically distributed NA random variables, and let be an array of constants satisfying
(17)
for some . Let for some . Furthermore, suppose that where . If
(18)
then
(19)

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.

Let be a sequence of identically distributed -mixing random variables, and let be an array of constants satisfying
(110)

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.

Under the conditions of Theorem 1.2, the assumptions for and (1.8) for imply the following Marcinkiewicz-Zygmund strong law:
(111)

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.

If is a sequence of random variables such that with and for every , then for all ,
(21)

where .

Lemma 2.2.

Let be a random variable and be an array of constants satisfying (1.10), . Then
(22)

Proof.

If , by and Lyapounov's inequality, then
(23)

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.

Let . For all , we have
(24)

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

By Lemma 2.2, one gets
(25)
Before the proof of , we prove firstly
(26)
For ,
(27)
For ,
(28)
Thus (2.6) holds. So, to prove , it is enough to show that
(29)
By the Chebyshev inequality and Lemma 2.1, for , we have
(210)

For , we consider the following two cases.

If , note that . We have
(211)

If , note that . we have

(212)

Next, we prove in the following two cases.

If or , take . Noting that , we have

(213)

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

(214)
for all . Hence, . Taking , we have
(215)

Proof of Theorem 1.4.

By (1.9), a standard computation (see page 120 of Baum and Katz [11] or page 1472 of An and Yuan [9]), and the Borel-Cantelli Lemma, we have
(216)
For any , there exists an integer such that . So
(217)
From (2.16) and (2.17), we have
(218)

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

(1)
Department of Mathematics, Southeast University, Nanjing, China
(2)
Department of Mathematics and Computer Science, Tongling University, Tongling, China
(3)
School of Mathematics, Heifei University of Technology, Hefei, China

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Copyright

© Xing-Cai Zhou et al. 2011

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.