Discrete Dynamics in Nature and Society

Volume 2012, Article ID 128492, 16 pages

http://dx.doi.org/10.1155/2012/128492

## Complete Convergence for Moving Average Process of Martingale Differences

School of Mathematical Science, Anhui University, Hefei 230039, China

Received 9 March 2012; Accepted 14 May 2012

Academic Editor: Chuanxi Qian

Copyright © 2012 Wenzhi Yang et al. 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.

#### Abstract

Under some simple conditions, by using some techniques such as truncated method for random variables (see e.g., Gut (2005)) and properties of martingale differences, we studied the moving process based on martingale differences and obtained complete convergence and complete moment convergence for this moving process. Our results extend some related ones.

#### 1. Introduction

Let be a doubly infinite sequence of random variables. Assume that is an absolutely summable sequence of real numbers and
is the *moving average process* based on the sequence . As usual, , , denotes the sequence of partial sums.

For the moving average process , where is a sequence of independent identically distributed (i.i.d.) random variables, Ibragimov [1] established the central limit theorem, Burton and Dehling [2] obtained a large deviation principle, and Li et al. [3] gave the complete convergence result for . Zhang [4] and Li and Zhang [5] extended the complete convergence of moving average process for i.i.d. sequence to -mixing sequence and NA sequence, respectively. Theorems A and B are due to Zhang [4] and Kim et al. [6], respectively.

Theorem A. *Suppose that is a sequence of identically distributed -mixing random variables with and is as in (1.1). Let be a slowly varying function and , . If and , then
*

Theorem B. *Suppose that is a sequence of identically distributed -mixing random variables with , and and is as in (1.1). Let be a slowly varying function and , . If , then
**
where .*

Chen et al. [7] and Zhou [8] also studied the limit behavior of moving average process under -mixing assumption. Yang et al. [9] investigated the moving average process for AANA sequence. For more works on complete convergence, one can refer to [3–6, 10–13] and the references therein.

Recall that the sequence is stochastically dominated by a nonnegative random variable if

Recently, Chen and Li [14] investigated the limit behavior of moving process under martingale difference sequences. They obtained the following theorems.

Theorem C. *Let , and . Assume that is a moving average process defined in (1.1), where is a martingale difference related to an increasing sequence of -fields and stochastically dominated by a nonnegative random variable . If , then for every ,
*

Theorem D. *Let , , and . Assume that is a moving average process defined in (1.1), where is a martingale difference related to an increasing sequence of -fields and stochastically dominated by a nonnegative random variable . If
**
then for every ,
**
where when and when and .*

Inspired by Chen and Li [14], Chen et al. [7], Sung [13] and other papers above, we go on to investigate the limit behavior of moving process under martingale difference sequence and obtain some similar results of Theorems C and D, but we only need some simple conditions. Our results extend some results of Chen and Li [14] (see Remark 3.3 in Section 3). Two lemmas and two theorems are given in Sections 2 and 3, respectively. The proofs of theorems are presented in Section 4.

For various results of martingales, one can refer to Chow [15], Hall and Heyde [16], Yu [17], Ghosal and Chandra [18], and so forth. As an application of moving average process based on martingale differences, we can refer to [19–22] and the references therein. Throughout the paper, is the indicator function of set , and , , denote some positive constants not depending on , which may be different in various places.

#### 2. Two Lemmas

The following lemmas are our basic techniques to prove our results.

Lemma 2.1 (cf. Hall and Heyde [16, Theorem 2.11]). * If is a martingale difference and , then there exists a constant depending only on such that
*

Lemma 2.2 (cf. Wu [23, Lemma 4.1.6]). * Let be a sequence of random variables, which is stochastically dominated by a nonnegative random variable . Then for any and , the following two statements hold:
**
where and are positive constants.*

#### 3. Main Results

Theorem 3.1. * Let and . Assume that is a moving average processes defined in (1.1), where is a martingale difference related to an increasing sequence of -fields and stochastically dominated by a nonnegative random variable . Let be a constant. Suppose that for and almost surely (a.s.), if . Then for every ,
*

Theorem 3.2. * Let the conditions of Theorem 3.1 hold. Then for every ,
*

*Remark 3.3. *Let be an increasing family of -algebras and be a sequence of martingale differences. Assume that for some ,
where is a constant not depending on , and other conditions are satisfied, Yu [17] investigated the complete convergence of weighted sums of martingale differences. On the other hand, under the condition
and other conditions, Ghosal and Chandra [18] obtained the complete convergence of martingale arrays. Thus, if , our assumption , ., is reasonable. Chen and Li [14] obtained Theorems C and D for the case . We go on to investigate this moving average process for the case , especially for the case and get the results of (3.1)–(3.4). If for , and , result (3.1) follows from Theorem C (see Theorem 1.1 of Chen and Li), but we can obtain results (3.1) and (3.2) under weaker condition . On the other hand, comparing with the conditions of Theorem D, our conditions of Theorem 3.2 are relatively simple.

#### 4. The Proofs of Main Results

*Proof of Theorem 3.1. *First, we show that the moving average process (1.1) converges a.s. under the conditions of Theorem 3.1. Since , it has , following from . On the other hand, applying Lemma 2.2 with and , one has
Consequently, we have by that
which implies converges a.s.

Note that
Let
Since , we can see that

For , by Markov’s inequality, Lemma 2.2, and , one has
Meanwhile, by the martingale property, Lemma 2.2 and the proof of (4.6), it follows that
Obviously, one can find that is a martingale difference. So, by Markov’s inequality, Hölder’s inequality, and Lemma 2.1, we get that for any ,
If , then we take large enough such that . From , a.s. and Jensen’s inequality for conditional expectation, we have , a.s. On the other hand,
Consequently, we obtain by that
following from the fact that . Meanwhile, by inequality, Lemma 2.2 and ,
Since and , one has
By the proof of (4.6),
If , then we take . Similar to the proofs of (4.8), and (4.11), it has
following from , (4.12), and (4.13). Therefore, (3.1) follows from (4.5)–(4.13) and the inequality above.

Inspired by the proof of Theorem 12.1 of Gut [24], it can be checked that
If , then
Otherwise,
Combining (3.1) with these inequalities above, we obtain (3.2) immediately.

*Proof of Theorem 3.2. *For all , it has
By Theorem 3.1, in order to proof (3.3), we only have to show that
For , denote
Since , it is easy to see that
By Markov’s inequality, Lemma 2.2 and ,
Since is a martingale difference, we have by Markov’s inequality, Hölder’s inequality, and Lemma 2.1 that for any ,
If , then we take large enough such that . By , a.s. and Jensen’s inequality for conditional expectation, it has , a.s.. Meanwhile,
Thus, by , one has that
following from the fact that . We also have by inequality and Lemma 2.2 that
Since and , it follows that
From the proof of (4.22),
If , then we take . Similar to the proofs of (4.23) and (4.26), we get that
following from , (4.27) and (4.28). Consequently, by (4.18)–(4.28), Theorem 3.1 and inequality above, (3.3) holds true.

Now, we turn to prove (3.4). Similar to the proof of (3.2), we have that
If , then
Otherwise,

Therefore, (3.4) holds true following from (3.3).

#### Acknowledgments

The authors are grateful to Editor Chuanxi Qian and an anonymous referee for their careful reading and insightful comments. This work is supported by the National Natural Science Foundation of China (11171001, 11126176), HSSPF of the Ministry of Education of China (10YJA910005), Natural Science Foundation of Anhui Province (1208085QA03), and Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005).

#### References

- I. A. Ibragimov, “Some limit theorems for stationary processes,”
*Theory of Probability and Its Applications*, vol. 7, no. 4, pp. 361–392, 1962. View at Google Scholar - R. M. Burton and H. Dehling, “Large deviations for some weakly dependent random processes,”
*Statistics & Probability Letters*, vol. 9, no. 5, pp. 397–401, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. L. Li, M. B. Rao, and X. C. Wang, “Complete convergence of moving average processes,”
*Statistics & Probability Letters*, vol. 14, no. 2, pp. 111–114, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. X. Zhang, “Complete convergence of moving average processes under dependence assumptions,”
*Statistics & Probability Letters*, vol. 30, no. 2, pp. 165–170, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. X. Li and L. X. Zhang, “Complete moment convergence of moving-average processes under dependence assumptions,”
*Statistics & Probability Letters*, vol. 70, no. 3, pp. 191–197, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T.-S. Kim, M.-H. Ko, and Y.-K. Choi, “Complete moment convergence of moving average processes with dependent innovations,”
*Journal of the Korean Mathematical Society*, vol. 45, no. 2, pp. 355–365, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Y. Chen, T.-C. Hu, and A. Volodin, “Limiting behaviour of moving average processes under
*ϕ*-mixing assumption,”*Statistics & Probability Letters*, vol. 79, no. 1, pp. 105–111, 2009. View at Publisher · View at Google Scholar - X. C. Zhou, “Complete moment convergence of moving average processes under
*ϕ*-mixing assumptions,”*Statistics & Probability Letters*, vol. 80, no. 5-6, pp. 285–292, 2010. View at Publisher · View at Google Scholar - W. Z. Yang, X. J. Wang, N. X. Ling, and S. H. Hu, “On complete convergence of moving average process for AANA sequence,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 863931, 24 pages, 2012. View at Publisher · View at Google Scholar - Y. S. Chow, “On the rate of moment convergence of sample sums and extremes,”
*Bulletin of the Institute of Mathematics*, vol. 16, no. 3, pp. 177–201, 1988. View at Google Scholar · View at Zentralblatt MATH - P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 33, pp. 25–31, 1947. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q. M. Shao, “A moment inequality and its applications,”
*Acta Mathematica Sinica*, vol. 31, no. 6, pp. 736–747, 1988. View at Google Scholar · View at Zentralblatt MATH - S. H. Sung, “Complete convergence for weighted sums of ${\rho}^{\ast}$-mixing random variables,”
*Discrete Dynamics in Nature and Society*, Article ID 630608, 13 pages, 2010. View at Publisher · View at Google Scholar - P. Y. Chen and Y. M. Li, “Limiting behavior of moving average processes of martingale difference sequences,”
*Acta Mathematica Scientia A*, vol. 31, no. 1, pp. 179–187, 2011. View at Google Scholar - Y. S. Chow, “Convergence of sums of squares of martingale differences,”
*Annals of Mathematical Statistics*, vol. 39, pp. 123–133, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Hall and C. C. Heyde,
*Martingale Limit Theory and Its Application*, Academic Press, New York, NY, USA, 1980. - K. F. Yu, “Complete convergence of weighted sums of martingale differences,”
*Journal of Theoretical Probability*, vol. 3, no. 2, pp. 339–347, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Ghosal and T. K. Chandra, “Complete convergence of martingale arrays,”
*Journal of Theoretical Probability*, vol. 11, no. 3, pp. 621–631, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. H. Hu, “Fixed-design semiparametric regression for linear time series,”
*Acta Mathematica Scientia B*, vol. 26, no. 1, pp. 74–82, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. H. Hu, X. Li, W. Yang, and X. Wang, “Maximal inequalities for some dependent sequences and their applications,”
*Journal of the Korean Statistical Society*, vol. 40, no. 1, pp. 11–19, 2011. View at Publisher · View at Google Scholar - S. H. Hu, C. H. Zhu, Y. B. Chen, and L. Wang, “Fixed-design regression for linear time series,”
*Acta Mathematica Scientia*, vol. 22, no. 1, pp. 9–18, 2002. View at Google Scholar - L. Tran, G. Roussas, S. Yakowitz, and B. Truong Van, “Fixed-design regression for linear time series,”
*The Annals of Statistics*, vol. 24, no. 3, pp. 975–991, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q. Y. Wu,
*Probability Limit Theory for Mixed Sequence*, Science Press, Beijing, China, 2006. - A. Gut,
*Probability: A Graduate Course*, Springer Science+Business Media,, New York, NY, USA, 2005.