Open Access

Some Strong Limit Theorems for Weighted Product Sums of -Mixing Sequences of Random Variables

Journal of Inequalities and Applications20092009:174768

DOI: 10.1155/2009/174768

Received: 4 June 2009

Accepted: 18 November 2009

Published: 19 November 2009

Abstract

We study almost sure convergence for -mixing sequences of random variables. Many of the previous results are our special cases. For example, the authors extend and improve the corresponding results of Chen et al. (1996) and Wu and Jiang (2008). We extend the classical Jamison convergence theorem and the Marcinkiewicz strong law of large numbers for independent sequences of random variables to -mixing sequences of random variables without necessarily adding any extra conditions.

1. Introduction and Lemmas

Let be a probability space. The random variables we deal with are all defined on . Let be a sequence of random variables. For each nonempty set , and write . Given -algebras in , let

(1.1)

where . Define the -mixing coefficients by

(1.2)

Obviously and except in the trivial case where all of the random variables are degenerate.

Definition 1.1.

A random variables sequence is said to be a -mixing random variables sequence if there exists such that .

-mixing is similar to -mixing, but both are quite different. A number of writers have studied -mixing random variables sequences and a series of useful results have been established. We refer to Bradley [1] (which assumes in the central limit theorem), Bryc and Smoleński [2], Goldie and Greenwood [3] (which assumes ), and Yang [4] for moment inequalities and the strong law of large numbers, Wu [5, 6], Wu and Jiang [7], Peligrad and Gut [8], and Gan [9] for almost sure convergence and Utev and Peligrad [10] for maximal inequalities and the invariance principle. When these are compared with the corresponding results of independent random variables sequences, there still remains much to be desired.

Lemma 1.2 (see [7, Theorem ]).

Let be a -mixing sequence of random variables which satisfies
(1.3)

Then converges almost surely (a.s.) and in quadratic mean.

Lemma 1.3 (see [11, Lemma ]).

For each positive integer , let denote the set of all vectors such that .

Then for each positive integer , there exists a function such that the following holds.

For any integer and any choice of real numbers , one has that

(1.4)

2. Main Results and the Proof

To state our results, we need some notions. Throughout this paper, let be a sequence of positive real numbers, and let satisfy .

Jamison et al. [12] proved the following result. Suppose that are i.i.d. random variables with . Denote , that is, the number of subscripts such that . If , then a.s. Chen et al. [13] extended the Jamison Theorem and obtained the following result. Suppose that are i.i.d. random variables with for some . If , then a.s.

The main purpose of this paper is to study the strong limit theorems for weighted sums of -mixing random variables sequences and try to obtain some new results. We establish weighted partial sums and weighted product sums strong convergence theorems. Our results in this paper extend and improve the corresponding results of Chen et al. [13], Wu and Jiang [7], the classical Jamison convergence theorem, and the Marcinkiewicz strong law of large numbers for independent sequences of random variables to -mixing sequences of random variables.

Theorem 2.1.

Let be a sequence of -mixing random variables with , and let the following conditions be satisfied:
(2.1)
(2.2)
(2.3)
where Then
(2.4)

Theorem 2.2.

Suppose that the assumptions of Theorem 2.1 hold, and also suppose
(2.5)
Then for all
(2.6)

Corollary 2.3.

Let be a sequence of -mixing identically distributed random variables. Let for some
(2.7)
(2.8)

Then (2.6) holds.

Remark 2.4.

Let be i.i.d. random variables, and in Corollary 2.3, then Corollary 2.3 is the well-known Jamison convergence theorem. Thus, our Theorem 2.2 and Corollary 2.3 generalize and improve the Jamison convergence theorem from the i.i.d. case to -mixing sequence. In addition, by Theorems and in Chen et al. [13] are special situation of Corollary 2.3.

Theorem 2.5.

Let be a sequence of -mixing random variables. Let be a sequence of positive real numbers with and let the following conditions be satisfied:
(2.9)
(2.10)
(2.11)
Then for all
(2.12)

Corollary 2.6.

Let be a -mixing identically distributed random variable sequence, for , and for . Then for all
(2.13)

In particular, taking , the above formula is the well-known Marcinkiewicz strong law of large numbers. Thus, our Theorem 2.5 and Corollary 2.6 generalize and improve the Marcinkiewicz strong law of large numbers from the i.i.d. case to -mixing sequence. In addition, by Theorem in Wu and Jiang [7] is a special case of Corollary 2.6.

Proof of Theorem 2.1.

Let . From (2.2),
(2.14)
By the Borel-Cantelli lemma and the Toeplitz lemma,
(2.15)
By and (2.1),
(2.16)
By (2.3),
(2.17)
Applying Lemma 1.2,
(2.18)
converges. Hence
(2.19)

from the Kronecker lemma. Combining (2.15)–(2.19), (2.4) holds. This completes the proof of Theorem 2.1.

Proof of Theorem 2.2.

By Lemma 1.3,
(2.20)
where denote the set of all vectors , such that , and are constants which do not depend on and Thus, in order to prove (2.6), we only need to prove that
(2.21)
When , by Theorem 2.1, (2.21) holds. When we get
(2.22)
from the elementary inequality valid for applied with . Hence, in order to prove (2.21), we only need to prove that
(2.23)
By (2.3), using Lemma 1.2, we get that
(2.24)
converges. By the Kronecker lemma, Thus,
(2.25)
that is,
(2.26)
By (2.3),
(2.27)
By Lemma 1.2, we have that
(2.28)
converges. By the Kronecker lemma,
(2.29)
By and the Toeplitz lemma,
(2.30)
Then combining (2.5), we obtain
(2.31)
Substituting (2.29) and (2.31) in (2.26), we get
(2.32)

Then combining (2.2) and the Borel-Cantelli lemma, (2.23) holds. This completes the proof of Theorem 2.2.

Proof of Corollary 2.3.

By Theorem 2.2, we only need to verify (2.1)–(2.3) and (2.5). From having identically distribution, (2.7), (2.8) and (2.5) holds automatically.

Since

(2.33)
by the Toeplitz lemma,
(2.34)

That is,(2.1) holds.

By (2.7),

(2.35)

That is, (2.2) holds.

Similarly,

(2.36)

That is, (2.3) holds. This completes proof of Corollary 2.3.

Proof of Theorem 2.5.

Similar to the proof of Theorem 2.2, by Lemma 1.3, in order to prove (2.12), we only need to prove that
(2.37)
Let then
(2.38)

 (i) When , by (2.10) and (2.11), in order to prove we only need to prove that

(2.39)
By (2.9) and Lemma 1.2,
(2.40)
converges. By the Kronecker lemma, (2.39) holds.
  1. (ii)

    When , by (2.9) and the Kronecker lemma,

     
(2.41)
By (2.10) and the Borel-Cantelli lemma,
(2.42)

Hence, combining (2.39), (2.37) holds. This completes the proof of Theorem 2.5.

Proof of Corollary 2.6.

Let . We can easy to verify (2.9)–(2.11). By Theorem 2.5, Corollary 2.6 holds.

Declarations

Acknowledgments

The authors are very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. This work was supported by the National Natural Science Foundation of China (10661006), the Support Program of the New Century Guangxi China Ten-hundred-thousand Talents Project (2005214), and the Guangxi, China Science Foundation (0991081).

Authors’ Affiliations

(1)
College of Science, Guilin University of Technology

References

  1. Bradley RC: On the spectral density and asymptotic normality of weakly dependent random fields. Journal of Theoretical Probability 1992,5(2):355–373. 10.1007/BF01046741MathSciNetView ArticleMATHGoogle Scholar
  2. Bryc W, Smoleński W: Moment conditions for almost sure convergence of weakly correlated random variables. Proceedings of the American Mathematical Society 1993,119(2):629–635. 10.1090/S0002-9939-1993-1149969-7MathSciNetView ArticleMATHGoogle Scholar
  3. Goldie CM, Greenwood PE: Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes. The Annals of Probability 1986,14(3):817–839. 10.1214/aop/1176992440MathSciNetView ArticleMATHGoogle Scholar
  4. Yang SC: Some moment inequalities for partial sums of random variables and their application. Chinese Science Bulletin 1998,43(17):1823–1826.MathSciNetView ArticleGoogle Scholar
  5. Wu QY: Some convergence properties for -mixing sequences. Journal of Engineering Mathematics 2001,18(3):58–64.MathSciNetMATHGoogle Scholar
  6. Wu QY: Convergence for weighted sums of -mixing random sequences. Mathematica Applicata 2002,15(1):1–4.MathSciNetMATHGoogle Scholar
  7. Wu QY, Jiang YY: Some strong limit theorems for -mixing sequences of random variables. Statistics & Probability Letters 2008,78(8):1017–1023. 10.1016/j.spl.2007.09.061MathSciNetView ArticleMATHGoogle Scholar
  8. Peligrad M, Gut A: Almost-sure results for a class of dependent random variables. Journal of Theoretical Probability 1999,12(1):87–104. 10.1023/A:1021744626773MathSciNetView ArticleMATHGoogle Scholar
  9. Gan SX: Almost sure convergence for -mixing random variable sequences. Statistics & Probability Letters 2004,67(4):289–298. 10.1016/j.spl.2003.12.011MathSciNetView ArticleMATHGoogle Scholar
  10. Utev S, Peligrad M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. Journal of Theoretical Probability 2003,16(1):101–115. 10.1023/A:1022278404634MathSciNetView ArticleMATHGoogle Scholar
  11. Wang YB, Yang Y, Zhou HY: A random functional central limit theorem for processes of product sums of linear processes generated by martingale differences. Chinese Annals of Mathematics B 2003,24(4):449–456. 10.1142/S025295990300044XMathSciNetView ArticleMATHGoogle Scholar
  12. Jamison B, Orey S, Pruitt W: Convergence of weighted averages of independent random variables. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1965, 4: 40–44. 10.1007/BF00535481MathSciNetView ArticleMATHGoogle Scholar
  13. Chen XR, Zhu L-C, Fang K-T: Almost sure convergence of weighted sums. Statistica Sinica 1996,6(2):499–507.MathSciNetMATHGoogle Scholar

Copyright

© Q.Wu and Y. Jiang. 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.