- Xing-cai Zhou
^{1, 2}and - Jin-guan Lin
^{1}Email author

**2010**:769201

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

© Xing-cai Zhou and Jin-guan Lin. 2010

**Received: **15 May 2010

**Accepted: **21 October 2010

**Published: **25 October 2010

## Abstract

## Keywords

## 1. Introduction

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

Definition 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.

Definition 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]).

Lemma 1.5 (Sung [19]).

Lemma 1.6 (Zhou [20]).

If is a slowly varying function as , then

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:

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.

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

## 3. Applications

Theorem 3.1.

Proof.

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

Theorem 3.2.

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

Theorem 3.3.

Proof.

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

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:

Proof.

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

Theorem 3.5.

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.

which proves that condition (A) is satisfied.

which proves that (B) holds.

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

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

## References

- Kolmogorov AN, Rozanov G: On the strong mixing conditions of a stationary Gaussian process.
*Theory of Probability and Its Applications*1960, 2: 222–227.MathSciNetMATHGoogle Scholar - Ibragimov IA: A note on the central limit theorem for dependent random variables.
*Theory of Probability and Its Applications*1975, 20: 134–139.Google Scholar - Peligrad M: On the central limit theorem for -mixing sequences of random variables.
*The Annals of Probability*1987, 15(4):1387–1394. 10.1214/aop/1176991983MathSciNetView ArticleMATHGoogle Scholar - Bradley RC: A central limit theorem for stationary -mixing sequences with infinite variance.
*The Annals of Probability*1988, 16(1):313–332. 10.1214/aop/1176991904MathSciNetView ArticleMATHGoogle Scholar - Peligrad M: Invariance principles for mixing sequences of random variables.
*The Annals of Probability*1982, 10(4):968–981. 10.1214/aop/1176993718MathSciNetView ArticleMATHGoogle Scholar - Shao QM: A remark on the invariance principle for -mixing sequences of random variables.
*Chinese Annals of Mathematics Series A*1988, 9(4):409–412.MathSciNetMATHGoogle Scholar - Shao QM: On the invariance principle for -mixing sequences of random variables.
*Chinese Annals of Mathematics Series B*1989, 10(4):427–433.MathSciNetMATHGoogle Scholar - Shao QM: Complete convergence of -mixing sequences.
*Acta Mathematica Sinica*1989, 32(3):377–393.MathSciNetMATHGoogle Scholar - 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 - Peligrad M: Convergence rates of the strong law for stationary mixing sequences.
*Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete*1985, 70(2):307–314. 10.1007/BF02451434MathSciNetView ArticleMATHGoogle Scholar - Shao QM: Maximal inequalities for partial sums of -mixing sequences.
*The Annals of Probability*1995, 23(2):948–965. 10.1214/aop/1176988297MathSciNetView ArticleMATHGoogle Scholar - Liang H, Yang C: A note of convergence rates for sums of -mixing sequences.
*Acta Mathematicae Applicatae Sinica*1999, 15(2):172–177. 10.1007/BF02720492MathSciNetView ArticleMATHGoogle Scholar - Hu T-C, Ordóñez Cabrera M, Sung SH, Volodin A: Complete convergence for arrays of rowwise independent random variables.
*Communications of the Korean Mathematical Society*2003, 18(2):375–383.View ArticleMathSciNetMATHGoogle Scholar - Sung SH, Volodin AI, Hu T-C: More on complete convergence for arrays.
*Statistics & Probability Letters*2005, 71(4):303–311. 10.1016/j.spl.2004.11.006MathSciNetView ArticleMATHGoogle Scholar - Kruglov VM, Volodin AI, Hu T-C: On complete convergence for arrays.
*Statistics & Probability Letters*2006, 76(15):1631–1640. 10.1016/j.spl.2006.04.006MathSciNetView ArticleMATHGoogle Scholar - Kuczmaszewska A: On complete convergence for arrays of rowwise dependent random variables.
*Statistics & Probability Letters*2007, 77(11):1050–1060. 10.1016/j.spl.2006.12.007MathSciNetView ArticleMATHGoogle Scholar - Kuczmaszewska A: On complete convergence for arrays of rowwise negatively associated random variables.
*Statistics & Probability Letters*2009, 79(1):116–124. 10.1016/j.spl.2008.07.030MathSciNetView ArticleMATHGoogle Scholar - Baek J-I, Park S-T: Convergence of weighted sums for arrays of negatively dependent random variables and its applications.
*Journal of Theoretical Probability*2010, 23(2):362–377. 10.1007/s10959-008-0198-yMathSciNetView ArticleMATHGoogle Scholar - Sung SH: Complete convergence for weighted sums of random variables.
*Statistics & Probability Letters*2007, 77(3):303–311. 10.1016/j.spl.2006.07.010MathSciNetView ArticleMATHGoogle Scholar - Zhou XC: Complete moment convergence of moving average processes under -mixing assumptions.
*Statistics & Probability Letters*2010, 80(5–6):285–292. 10.1016/j.spl.2009.10.018MathSciNetView ArticleMATHGoogle Scholar

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