Skip to content

Advertisement

Open Access

On Complete Convergence for Weighted Sums of -Mixing Random Variables

Journal of Inequalities and Applications20102010:372390

https://doi.org/10.1155/2010/372390

Received: 4 February 2010

Accepted: 11 June 2010

Published: 6 July 2010

Abstract

Some results on complete convergence for weighted sums are presented, where , is a sequence of -mixing random variables and is an array of constants. They generalize the corresponding results for sequence to the case of -mixing sequence.

Keywords

Positive IntegerReal NumberLimit TheoremProbability SpaceCentral Limit

1. Introduction

Let , be a sequence of random variables defined on a fixed probability space . 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 .

-mixing random variables were 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, Shao [8] for almost sure invariance principles, Hu and Wang [9] 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.

Throughout the paper, let be the indicator function of the set . We assume that is a positive increasing function on satisfying as and is the inverse function of . Since , it follows that . For easy notation, we let and . denotes that there exists a positive constant such that . denotes a positive constant which may be different in various places.

Let , be a sequence of random variables and let , be an array of constants. The almost sure limiting behavior of weighted sums was studied by many authors; see, for example, Choi and Sung [10], Cuzick [11], Wu [12], and Sung [13, 14], and so forth.

The main purpose of this paper is to extend the complete convergence for weighted sums of random variables to the case of -mixing random variables.

Definition 1.2.

A sequence of random variables is said to be stochastically dominated by a random variable if there exists a positive constant , such that
(1.3)

for all and .

Definition 1.3.

A double array , , of real numbers is said to be a Toeplitz array if for each and
(1.4)

for all , where is a positive constant.

Lemma 1.4.

Let , be a sequence of random variables which is stochastically dominated by a random variable . For any and , the following statement holds:
(1.5)

where is a positive constant.

Lemma 1.5 (cf. [15, Lemma ]).

Let , be a sequence of -mixing random variables. Let , , , , and . Then
(1.6)

Lemma 1.6 (cf. [8, Lemma 2.2]).

Let , be a -mixing sequence. Put . Suppose that there exists an array of positive numbers such that
(1.7)
Then for every , there exists a constant depending only on and such that
(1.8)

for every and .

Lemma 1.7.

Let , be a sequence of -mixing random variables satisfying . . Assume that and for each . Then there exists a constant depending only on and such that
(1.9)
for every and . In particular, one has
(1.10)

for every .

Proof.

By Lemma 1.5, we can see that
(1.11)

which implies (1.7). By Lemma 1.6, we can get the desired result (1.9) immediately. The proof is complete.

Lemma 1.8.

Assume that the inverse function of satisfies
(1.12)
If , then
(1.13)

Proof.

The proof is similar to that of Lemma by Sung [14]. So we omit it.

2. Main Results and Their Proofs

Theorem 2.1.

Let , be a sequence of identically distributed -mixing random variables satisfying , , , and . Assume that the inverse function of satisfies (1.12). Let , , be an array of constants such that

(i) ;

(ii) for some .

Then for any ,
(2.1)

Proof.

For each , denote
(2.2)
It is easy to check that
(2.3)
Therefore
(2.4)
Firstly, we will show that
(2.5)
It follows from Lemma 1.8 and Kronecker's lemma that
(2.6)
By , condition (i), (2.6), and , we can see that
(2.7)
which implies (2.5). By (2.4) and (2.5), we can see that, for sufficiently large ,
(2.8)
To prove (2.1), it suffices to show that
(2.9)
By Markov's inequality, Lemma 1.7, , and condition (ii), we have
(2.10)
It follows from that
(2.11)

We complete the proof of the theorem.

Theorem 2.2.

Let , be a sequence of -mixing random variables satisfying and let , , be an array of real numbers. Let , be an increasing sequence of positive integers and let , be a sequence of positive real numbers. If for some , , and for any , the following conditions are satisfied:
(2.12)
(2.13)
(2.14)
then
(2.15)

Proof.

Note that if the series is convergent, then (2.15) holds. Therefore, we will consider only such sequences , for which the series is divergent.

Let
(2.16)
Therefore
(2.17)
Using the inequality and Jensen's inequality, we can estimate in the following way:
(2.18)
By (2.17), (2.18), and Lemma 1.7, we can get
(2.19)

Therefore, we can conclude that (2.15) holds by (2.12), (2.13), (2.14), and (2.19).

Theorem 2.3.

Let and let , be a sequence of -mixing random variables satisfying , , and for . Let , , be an array of real numbers satisfying the following condition:
(2.20)
for some and . Then for any and ,
(2.21)

Proof.

Take , , and in Theorem 2.2. By (2.20) we have
(2.22)
following from . By the assumption for and (2.20) we get
(2.23)

following from and . We get the desired result by Theorem 2.2 immediately. The proof is completed.

Theorem 2.4.

Let and let , be a sequence of -mixing random variables satisfying , , and for . Assume that the random variables are stochastically dominated by a random variable such that and let , , be an array of real numbers satisfying the following condition:
(2.24)

for some and . Then for any and , (2.21) holds.

Proof.

The proof is similar to that of Theorem 2.3. We only need to note that
(2.25)

for each .

Theorem 2.5.

Let , be a sequence of -mixing random variables satisfying and let , , be a Toeplitz array. Assume that the random variables are stochastically dominated by a random variable . If for some and ,
(2.26)
where , then for any ,
(2.27)

Proof.

Take , for and in Theorem 2.2. Then we can see that (2.12) and (2.13) are satisfied. In fact, by (1.4) and (2.26) we have
(2.28)
and by Lemma 1.4, (1.5), and (2.26) we have
(2.29)

In order to prove that (2.14) holds, we should consider the following two cases.

In the case , by Lemma 1.4, (1.5), (2.26), and inequality, we have
(2.30)
In the case , we can get
(2.31)
To complete the proof of the theorem, we only need to prove
(2.32)
Indeed, by Lemma 1.4, it follows that
(2.33)

Thus we get the desired result.

Declarations

Acknowledgments

The authors are most grateful to the Editor Andrei I. Volodin and anonymous referee for the careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (10871001, 60803059), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Talents Youth Fund of Anhui Province Universities (2010SQRL016ZD), and Youth Science Research Fund of Anhui University (2009QN011A).

Authors’ Affiliations

(1)
School of Mathematical Science, Anhui University, Hefei, China

References

  1. Dobrushin RL: The central limit theorem for non-stationary Markov chain. Theory of Probability and Its Applications 1956, 1: 72–88.MathSciNetMATHGoogle Scholar
  2. Utev SA: The central limit theorem for -mixing arrays of random variables. Theory of Probability and Its Applications 1990, 35(1):131–139. 10.1137/1135013MathSciNetView ArticleMATHGoogle Scholar
  3. Chen DC: A uniform central limit theorem for nonuniform -mixing random fields. The Annals of Probability 1991, 19(2):636–649. 10.1214/aop/1176990445MathSciNetView ArticleMATHGoogle Scholar
  4. Herrndorf N: The invariance principle for -mixing sequences. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1983, 63(1):97–108. 10.1007/BF00534180MathSciNetView ArticleMATHGoogle Scholar
  5. Peligrad M: An invariance principle for -mixing sequences. The Annals of Probability 1985, 13(4):1304–1313. 10.1214/aop/1176992814MathSciNetView ArticleMATHGoogle Scholar
  6. Sen PK: A note on weak convergence of empirical processes for sequences of -mixing random variables. Annals of Mathematical Statistics 1971, 42: 2131–2133. 10.1214/aoms/1177693079MathSciNetView ArticleMATHGoogle Scholar
  7. Sen PK: Weak convergence of multidimensional empirical processes for stationary -mixing processes. The Annals of Probability 1974, 2(1):147–154. 10.1214/aop/1176996760View ArticleMathSciNetMATHGoogle Scholar
  8. Shao QM: Almost sure invariance principles for mixing sequences of random variables. Stochastic Processes and Their Applications 1993, 48(2):319–334. 10.1016/0304-4149(93)90051-5MathSciNetView ArticleMATHGoogle Scholar
  9. Hu S, Wang X: Large deviations for some dependent sequences. Acta Mathematica Scientia. Series B 2008, 28(2):295–300. 10.1016/S0252-9602(08)60030-2MathSciNetView ArticleMATHGoogle Scholar
  10. Choi BD, Sung SH: Almost sure convergence theorems of weighted sums of random variables. Stochastic Analysis and Applications 1987, 5(4):365–377. 10.1080/07362998708809124MathSciNetView ArticleMATHGoogle Scholar
  11. Cuzick J: A strong law for weighted sums of i.i.d. random variables. Journal of Theoretical Probability 1995, 8(3):625–641. 10.1007/BF02218047MathSciNetView ArticleMATHGoogle Scholar
  12. Wu WB: On the strong convergence of a weighted sum. Statistics & Probability Letters 1999, 44(1):19–22. 10.1016/S0167-7152(98)00287-9MathSciNetView ArticleMATHGoogle Scholar
  13. Sung SH: Strong laws for weighted sums of i.i.d. random variables. Statistics & Probability Letters 2001, 52(4):413–419. 10.1016/S0167-7152(01)00020-7MathSciNetView ArticleMATHGoogle Scholar
  14. Sung SH: Strong laws for weighted sums of i.i.d. random variables. II. Bulletin of the Korean Mathematical Society 2002, 39(4):607–615.MathSciNetView ArticleMATHGoogle Scholar
  15. Lu CR, Lin ZY: Limit Theory for Mixing Dependent Sequences. Science Press, Beijing, China; 1997.Google Scholar

Copyright

© Wang Xuejun et al. 2010

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.

Advertisement