- Research Article
- Open access
- Published:
On Complete Convergence for Arrays of Rowwise -Mixing Random Variables and Its Applications
Journal of Inequalities and Applications volume 2010, Article number: 769201 (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.
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
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
for all , , and .
Definition 1.3.
A real-valued function , positive and measurable on for some , is said to be slowly varying if
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
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:
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
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
Note that
By (a) it is enough to prove that for all
By Markov inequality and Lemma 1.4, and note that the assumption for some , we get
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
for some . Then for any and
Proof.
Put , , and in Theorem 2.1. By (3.1), we get
following from . By the assumption for , and by (3.1), we have
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
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 ,
then for any
Proof.
Take and for . Then we see that (a) and (b) are satisfied. Indeed, taking , by Lemma 1.5 and (3.6), we get
In order to prove that (c) holds, we consider the following two cases.
If , by Lemma 1.5, inequality, and (3.6), we have
If , take . We have that . Note that in this case . We have
The proof will be completed if we show that
Indeed, by Lemma 1.5, we have
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
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
then
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
which proves that condition (A) is satisfied.
Taking , we have . By Lemma 1.6, we have
which proves that (B) holds.
In order to prove that (C) holds, we consider the following two cases.
If , take . We have
If , take . We have . Note that in this case . We obtain
The proof will be completed if we show that
If , then
If , note that , then
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.
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.
Ibragimov IA: A note on the central limit theorem for dependent random variables. Theory of Probability and Its Applications 1975, 20: 134–139.
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/1176991983
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/1176991904
Peligrad M: Invariance principles for mixing sequences of random variables. The Annals of Probability 1982, 10(4):968–981. 10.1214/aop/1176993718
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.
Shao QM: On the invariance principle for -mixing sequences of random variables. Chinese Annals of Mathematics Series B 1989, 10(4):427–433.
Shao QM: Complete convergence of -mixing sequences. Acta Mathematica Sinica 1989, 32(3):377–393.
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)90051-5
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/BF02451434
Shao QM: Maximal inequalities for partial sums of -mixing sequences. The Annals of Probability 1995, 23(2):948–965. 10.1214/aop/1176988297
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/BF02720492
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.
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.2004.11.006
Kruglov VM, Volodin AI, Hu T-C: On complete convergence for arrays. Statistics & Probability Letters 2006, 76(15):1631–1640. 10.1016/j.spl.2006.04.006
Kuczmaszewska A: On complete convergence for arrays of rowwise dependent random variables. Statistics & Probability Letters 2007, 77(11):1050–1060. 10.1016/j.spl.2006.12.007
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.2008.07.030
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-y
Sung SH: Complete convergence for weighted sums of random variables. Statistics & Probability Letters 2007, 77(3):303–311. 10.1016/j.spl.2006.07.010
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.2009.10.018
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zhou, Xc., Lin, Jg. On Complete Convergence for Arrays of Rowwise -Mixing Random Variables and Its Applications. J Inequal Appl 2010, 769201 (2010). https://doi.org/10.1155/2010/769201
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/769201