# On Complete Convergence for Arrays of Rowwise Open image in new window -Mixing Random Variables and Its Applications

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## Abstract

We give out a general method to prove the complete convergence for arrays of rowwise Open image in new window -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 Number Positive Constant Convergence Rate Limit Theorem Central Limit## 1. Introduction

Let Open image in new window be a probability space, and let Open image in new window be a sequence of random variables defined on this space.

Definition 1.1.

as Open image in new window , where Open image in new window denotes the Open image in new window -field generated by Open image in new window .

The Open image in new window -mixing random variables were first introduced by Kolmogorov and Rozanov [1]. The limiting behavior of Open image in new window -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 Open image in new window -mixing and Open image in new window -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 Open image in new window -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.

for all Open image in new window , Open image in new window , and Open image in new window .

Definition 1.3.

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

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 Open image in new window is a slowly varying function as Open image in new window , then

(i) Open image in new window for Open image in new window ,

(ii) Open image in new window for Open image in new window .

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 Open image in new window -mixing sequence. Let Open image in new window be the mixing coefficient defined in Definition 1.1 for the Open image in new window th row of an array Open image in new window , that is, for the sequence Open image in new window .

Now, we state our main result.

Theorem 2.1.

Let Open image in new window be an array of rowwise Open image in new window -mixing random variables satisfying Open image in new window for some Open image in new window , and let Open image in new window be an array of real numbers. Let Open image in new window be an increasing sequence of positive integers, and let Open image in new window be a sequence of positive real numbers. If for some Open image in new window and any Open image in new window the following conditions are fulfilled:

(a) Open image in new window ,

(b) Open image in new window ,

(c) Open image in new window ,

Remark 2.2.

Theorem 2.1 extends some results of Kuczmaszewska [17] to the case of arrays of rowwise Open image in new window -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]: Open image in new window is only required in Theorem 2.1, not Open image in new window in Theorem 2.1 of Kuczmaszewska [17]; the powers of Open image in new window in (b) and (c) of Theorem 2.1 are Open image in new window and Open image in new window , respectively, not Open image in new window 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 Open image in new window and Open image in new window . Thus, we complete the proof of the theorem.

Theorem 3.2.

for some Open image in new window . Then for any Open image in new window and Open image in new window (3.2) holds.

Theorem 3.3.

Proof.

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

Theorem 3.4.

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

Proof.

Let Open image in new window and Open image in new window . Using Theorem 2.1, we obtain (3.13) easily.

Theorem 3.5.

Proof.

Put Open image in new window and Open image in new window for Open image in new window , Open image in new window 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, Open image in new window and Open image in new window , we can get the simpler formulas in the above theorems.

## Notes

### 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).

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