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On Complete Convergence for Arrays of Rowwise -Mixing Random Variables and Its Applications

Journal of Inequalities and Applications20102010:769201

https://doi.org/10.1155/2010/769201

Received: 15 May 2010

Accepted: 21 October 2010

Published: 25 October 2010

Abstract

We give out a general method to prove the complete convergence for arrays of rowwise -mixing random variables and to present some results on complete convergence under some suitable conditions. Some results generalize previous known results for rowwise independent random variables.

Keywords

Real NumberPositive ConstantConvergence RateLimit TheoremCentral Limit

1. Introduction

Let be a probability space, and let be a sequence of random variables defined on this space.

Definition 1.1.

The sequence is said to be -mixing if
(1.1)

as , where denotes the -field generated by .

The -mixing random variables were first introduced by Kolmogorov and Rozanov [1]. The limiting behavior of -mixing random variables is very rich, for example, these in the study by Ibragimov [2], Peligrad [3], and Bradley [4] for central limit theorem; Peligrad [5] and Shao [6, 7] for weak invariance principle; Shao [8] for complete convergence; Shao [9] for almost sure invariance principle; Peligrad [10], Shao [11] and Liang and Yang [12] for convergence rate; Shao [11], for the maximal inequality, and so forth.

For arrays of rowwise independent random variables, complete convergence has been extensively investigated (see, e.g., Hu et al. [13], Sung et al. [14], and Kruglov et al. [15]). Recently, complete convergence for arrays of rowwise dependent random variables has been considered. We refer to Kuczmaszewska [16] for -mixing and -mixing sequences, Kuczmaszewska [17] for negatively associated sequence, and Baek and Park [18] for negatively dependent sequence. In the paper, we study the complete convergence for arrays of rowwise -mixing sequence under some suitable conditions using the techniques of Kuczmaszewska [16, 17]. We consider the case of complete convergence of maximum weighted sums, which is different from Kuczmaszewska [16]. Some results also generalize some previous known results for rowwise independent random variables.

Now, we present a few definitions needed in the coming part of this paper.

Definition 1.2.

An array of random variables is said to be stochastically dominated by a random variable if there exists a constant , such that
(1.2)

for all , , and .

Definition 1.3.

A real-valued function , positive and measurable on for some , is said to be slowly varying if
(1.3)

Throughout the sequel, will represent a positive constant although its value may change from one appearance to the next; indicates the maximum integer not larger than ; denotes the indicator function of the set .

The following lemmas will be useful in our study.

Lemma 1.4 (Shao [11]).

Let be a sequence of -mixing random variables with and for some . Then there exists a positive constant depending only on and such that for any
(1.4)

Lemma 1.5 (Sung [19]).

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

Lemma 1.6 (Zhou [20]).

If is a slowly varying function as , then

(i) for ,

(ii) for .

This paper is organized as follows. In Section 2, we give the main result and its proof. A few applications of the main result are provided in Section 3.

2. Main Result and Its Proof

This paper studies arrays of rowwise -mixing sequence. Let be the mixing coefficient defined in Definition 1.1 for the th row of an array , that is, for the sequence .

Now, we state our main result.

Theorem 2.1.

Let be an array of rowwise -mixing random variables satisfying for some , 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 any the following conditions are fulfilled:

(a) ,

(b) ,

(c) ,

then
(2.1)

Remark 2.2.

Theorem 2.1 extends some results of Kuczmaszewska [17] to the case of arrays of rowwise -mixing sequence and generalizes the results of Kuczmaszewska [16] to the case of maximum weighted sums.

Remark 2.3.

Theorem 2.1 firstly gives the condition of the mixing coefficient, so the conditions (a)–(c) do not contain the mixing coefficient. Thus, the conditions (a)–(c) are obviously simpler than the conditions (i)–(iii) in Theorem  2.1 of Kuczmaszewska [16]. Our conditions are also different from those of Theorem  2.1 in the study by Kuczmaszewska [17]: is only required in Theorem 2.1, not in Theorem  2.1 of Kuczmaszewska [17]; the powers of in (b) and (c) of Theorem 2.1 are and , respectively, not in Theorem  2.1 of Kuczmaszewska [17].

Now, we give the proof of Theorem 2.1.

Proof.

The conclusion of the theorem is obvious if is convergent. Therefore, we will consider that only is divergent. Let
(2.2)
Note that
(2.3)
By (a) it is enough to prove that for all
(2.4)
By Markov inequality and Lemma 1.4, and note that the assumption for some , we get
(2.5)

From (b), (c), and (2.5), we see that (2.4) holds.

3. Applications

Theorem 3.1.

Let be an array of rowwise -mixing random variables satisfying for some , , and for all , , and . Let be an array of real numbers satisfying the condition
(3.1)
for some . Then for any and
(3.2)

Proof.

Put , , and in Theorem 2.1. By (3.1), we get
(3.3)
following from . By the assumption for , and by (3.1), we have
(3.4)

because and . Thus, we complete the proof of the theorem.

Theorem 3.2.

Let be an array of rowwise -mixing random variables satisfying for some , , and for all , , and . Let the random variables in each row be stochastically dominated by a random variable , such that , and let be an array of real numbers satisfying the condition
(3.5)

for some . Then for any and (3.2) holds.

Theorem 3.3.

Let be an array of rowwise -mixing random variables satisfying for some and for all , . Let the random variables in each row be stochastically dominated by a random variable , and let be an array of real numbers. If for some ,
(3.6)
then for any
(3.7)

Proof.

Take and for . Then we see that (a) and (b) are satisfied. Indeed, taking , by Lemma 1.5 and (3.6), we get
(3.8)

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

If , by Lemma 1.5, inequality, and (3.6), we have
(3.9)
If , take . We have that . Note that in this case . We have
(3.10)
The proof will be completed if we show that
(3.11)
Indeed, by Lemma 1.5, we have
(3.12)

Theorem 3.4.

Let be an array of rowwise -mixing random variables satisfying for some , and let be an array of real numbers. Let be a slowly varying function as . If for some and real number , and any the following conditions are fulfilled:

,

,

,

then
(3.13)

Proof.

Let and . Using Theorem 2.1, we obtain (3.13) easily.

Theorem 3.5.

Let be an array of rowwise -mixing identically distributed random variables satisfying for some and . Let be a slowly varying function as . If for , , and
(3.14)
then
(3.15)

Proof.

Put and for , in Theorem 3.4. To prove (3.15), it is enough to note that under the assumptions of Theorem 3.4, the conditions (A)–(C) of Theorem 3.4 hold.

By Lemma 1.6, we obtain
(3.16)

which proves that condition (A) is satisfied.

Taking , we have . By Lemma 1.6, we have
(3.17)

which proves that (B) holds.

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

If , take . We have
(3.18)
If , take . We have . Note that in this case . We obtain
(3.19)
The proof will be completed if we show that
(3.20)
If , then
(3.21)
If , note that , then
(3.22)

We complete the proof of the theorem.

Noting that for typical slowly varying functions, and , we can get the simpler formulas in the above theorems.

Declarations

Acknowledgments

The authors thank the academic editor and the reviewers for comments that greatly improved the paper. This work is partially supported by the Anhui Province College Excellent Young Talents Fund Project of China (no. 2009SQRZ176ZD) and National Natural Science Foundation of China (nos. 11001052, 10871001, 10971097).

Authors’ Affiliations

(1)
Department of Mathematics, Southeast University, Nanjing, China
(2)
Department of Mathematics and Computer Science, Tongling University, Tongling, China

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Copyright

© Xing-cai Zhou and Jin-guan Lin. 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.

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