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

  • Xing-cai Zhou
  • Jin-guan Lin
Open Access
Research Article
  • 926 Downloads

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

An array Open image in new window of random variables is said to be stochastically dominated by a random variable Open image in new window if there exists a constant Open image in new window , such that

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

Definition 1.3.

A real-valued function Open image in new window , positive and measurable on Open image in new window for some Open image in new window , is said to be slowly varying if

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]).

Let Open image in new window be a sequence of random variables which is stochastically dominated by a random variable Open image in new window . For any Open image in new window and Open image in new window , the following statement holds:

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.

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

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.

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 , Open image in new window , and Open image in new window for all Open image in new window , Open image in new window , and Open image in new window . Let the random variables in each row be stochastically dominated by a random variable Open image in new window , such that Open image in new window , and let Open image in new window be an array of real numbers satisfying the condition

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.

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 Open image in new window for all Open image in new window , Open image in new window . Let the random variables in each row be stochastically dominated by a random variable Open image in new window , and let Open image in new window be an array of real numbers. If for some Open image in new window , Open image in new window

Proof.

Take Open image in new window and Open image in new window for Open image in new window . Then we see that (a) and (b) are satisfied. Indeed, taking Open image in new window , by Lemma 1.5 and (3.6), we get

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

If Open image in new window , by Lemma 1.5, Open image in new window inequality, and (3.6), we have
The proof will be completed if we show that
Indeed, by Lemma 1.5, we have

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:

Open image in new window ,

Open image in new window ,

Open image in new window ,

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.

By Lemma 1.6, we obtain

which proves that condition (A) is satisfied.

Taking Open image in new window , we have Open image in new window . By Lemma 1.6, we have

which proves that (B) holds.

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

The proof will be completed if we show that

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).

References

  1. 1.
    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
  2. 2.
    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
  3. 3.
    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/1176991983MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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/1176991904MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Peligrad M: Invariance principles for mixing sequences of random variables. The Annals of Probability 1982, 10(4):968–981. 10.1214/aop/1176993718MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    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
  8. 8.
    Shao QM: Complete convergence of -mixing sequences. Acta Mathematica Sinica 1989, 32(3):377–393.MathSciNetMATHGoogle Scholar
  9. 9.
    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)90051-5MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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/BF02451434MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Shao QM: Maximal inequalities for partial sums of -mixing sequences. The Annals of Probability 1995, 23(2):948–965. 10.1214/aop/1176988297MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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/BF02720492MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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.CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    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.2004.11.006MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kruglov VM, Volodin AI, Hu T-C: On complete convergence for arrays. Statistics & Probability Letters 2006, 76(15):1631–1640. 10.1016/j.spl.2006.04.006MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kuczmaszewska A: On complete convergence for arrays of rowwise dependent random variables. Statistics & Probability Letters 2007, 77(11):1050–1060. 10.1016/j.spl.2006.12.007MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    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.2008.07.030MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    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-yMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sung SH: Complete convergence for weighted sums of random variables. Statistics & Probability Letters 2007, 77(3):303–311. 10.1016/j.spl.2006.07.010MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    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.2009.10.018MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© 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.

Authors and Affiliations

  1. 1.Department of MathematicsSoutheast UniversityNanjingChina
  2. 2.Department of Mathematics and Computer ScienceTongling UniversityTonglingChina

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