Open Access

Precise Asymptotics in the Law of Iterated Logarithm for Moving Average Process under Dependence

Journal of Inequalities and Applications20112011:320932

https://doi.org/10.1155/2011/320932

Received: 10 November 2010

Accepted: 3 March 2011

Published: 15 March 2011

Abstract

Let be a doubly infinite sequence of identically distributed and -mixing random variables, and let be an absolutely summable sequence of real numbers. In this paper, we get precise asymptotics in the law of the logarithm for linear process , , which extend Liu and Lin's (2006) result to moving average process under dependence assumption.

1. Introduction and Main Results

Let be a doubly infinite sequence of identically distributed random variables with zero means and finite variances, and let be an absolutely summable sequence of real numbers. Let
(11)

be the moving average process based on . As usual, we denote , as the sequence of partial sums.

Under the assumption that is a sequence of independent identically distributed random variables, many limiting results have been obtained. Ibragimov [1] established the central limit theorem; Burton and Dehling [2] obtained a large deviation principle; Yang [3] established the central limit theorem and the law of the iterated logarithm; Li et al. [4] obtained the complete convergence result for . As we know, are dependent even if is a sequence of i.i.d. random variables. Therefore, we introduce the definition of -mixing,
(12)

where . Many limiting results of moving average for -mixing have been obtained. For example, Zhang [5] got complete convergence.

Theorem A.

Suppose that is a sequence of identically distributed and -mixing random variables with , and is defined as (1.1). Let be a slowly varying function and , , then and imply
(13)

Li and Zhang [6] achieved precise asymptotics in the law of the iterated logarithm.

Theorem B.

Suppose that is a sequence of identically distributed and -mixing random variables with mean zeros and finite variances, , and , , for . Suppose that is defined as in (1.1), where is a sequence of real number with , then one has
(14)

where , is a standard normal random variable.

On the other hand, since Hsu and Robbins [7] introduced the concept of the complete convergence, there have been extensions in some directions. For the case of i.i.d. random variables, Davis [8] proved , for if and only if . Gut and Spătaru [9] gave the precise asymptotics of . We know that complete convergence can be derived from complete moment convergence. Liu and Lin [10] introduced a new kind of convergence of . In this note, we show that the precise asymptotics for the moment convergence hold for moving-average process when is a strictly stationary -mixing sequences. Now, we state the main results.

Theorem 1.1.

Suppose that is defined as in (1.1), where is a sequence of real number with , and is a sequence of identically distributed -mixing random variables with mean zeros and finite variances, and , , for , then one has
(15)

where .

Theorem 1.2.

Under the conditions in Theorem 1.1, one has
(16)

Remark 1.3.

In this paper, we generate the results of Liu and Lin [10] to linear process under dependence based on Theorem B by using the technique of dealing with the innovation process in Zhang [5].

We first proceed with some useful lemmas.

Lemma 1.4.

Let be defined as in (1.1), and let be a sequence of identically distributed -mixing random variables with , , , then
(17)

The proof is similar to Theorem 1 in [11]. Set . From Lemma 1.4, one can get as .

Lemma 1.5 (see [2]).

Let be an absolutely convergent series of real numbers with and , then
(18)

Lemma 1.6 (see [12]).

Let be a sequence of -mixing random variables with zero means and finite second moments. Let . If exists such that , then for all , there exists such that
(19)

2. Proofs

Proof of Theorem 1.1.

Without loss of generality, we assume that . We have
(21)
Set , where . By Theorem B, we need to show
(22)
By Proposition 5.1 in [10], we have
(23)

Hence, Theorem 1.1 will be proved if we show the following two propositions.

Proposition 2.1.

One has
(24)

Proof.

Write
(25)
where
(26)
Since implies , we have
(27)
For , by Markov's inequality, we get
(28)
From (2.7) and (2.8), we can get
(29)
Note that , where . By Lemma 1.5, we can assume that
(210)
Set . As , by (2.10), we have
(211)
So, when ,
(212)
By (2.12), we have
(213)
Set , then (referred by [4]). We can get
(214)
Then,
(215)
So, we get
(216)
Therefore,
(217)
By Lemma 1.6, noting that , for ,
(218)
For , we have
(219)
Then, for , , we have
(220)
For , we decompose it into two parts,
(221)
It is easy to see that
(222)
So,
(223)
Now, we estimate , by (2.23),
(224)
For , we have
(225)
From (2.24) and (2.25), we can get
(226)
Finally, , and we will get
(227)
then
(228)

Hence, (2.4) can be referred from (2.9), (2.17), (2.20), (2.26), and (2.28).

Proposition 2.2.

One has
(229)

Proof.

Consider the following:
(230)
We first estimate , for , by Markov's inequality,
(231)
Hence,
(232)
Now, we estimate . Here, , so
(233)
We have
(234)
We estimate first. Similar to the proof of (2.16), we have
(235)
then
(236)
By Lemma 1.6, for , we have
(237)
For , we have
(238)
Next, turning to , it follows that
(239)
then
(240)
For , it follows that
(241)
Finally, , we have
(242)
From (2.38) to (2.42), we can get
(243)

(2.29) can be derived by (2.32), (2.36), and (2.43).

Proof of Theorem 1.2.

Without loss of generality, we set . It is easy to see that
(244)
So, we only prove the following two propositions:
(245)
(246)

The proof of (2.45) can be referred to [6], and the proof of (2.46) is similar to Propositions 2.1 and 2.2.

Declarations

Acknowledgments

The author would like to thank the referee for many valuable comments. This research was supported by Humanities and Social Sciences Planning Fund of the Ministry of Education of PRC. (no. 08JA790118 )

Authors’ Affiliations

(1)
Wang Yanan Institute for Studies in Economics, Xiamen University
(2)
School of Mathematics and Statistics, Zhejiang University of Finance and Economics

References

  1. Ibragimov IA: Some limit theorems for stationary processes. Theory of Probability and Its Applications 1962, 7: 349–382. 10.1137/1107036View ArticleGoogle Scholar
  2. Burton RM, Dehling H: Large deviations for some weakly dependent random processes. Statistics & Probability Letters 1990,9(5):397–401. 10.1016/0167-7152(90)90031-2MATHMathSciNetView ArticleGoogle Scholar
  3. Yang XY: The law of the iterated logarithm and the central limit theorem with random indices for B-valued stationary linear processes. Chinese Annals of Mathematics Series A 1996,17(6):703–714.MATHMathSciNetGoogle Scholar
  4. Li DL, Rao MB, Wang XC: Complete convergence of moving average processes. Statistics & Probability Letters 1992,14(2):111–114. 10.1016/0167-7152(92)90073-EMATHMathSciNetView ArticleGoogle Scholar
  5. Zhang L-X: Complete convergence of moving average processes under dependence assumptions. Statistics & Probability Letters 1996,30(2):165–170. 10.1016/0167-7152(95)00215-4MATHMathSciNetView ArticleGoogle Scholar
  6. Li YX, Zhang LX: Precise asymptotics in the law of the iterated logarithm of moving-average processes. Acta Mathematica Sinica (English Series) 2006,22(1):143–156. 10.1007/s10114-005-0542-4MATHMathSciNetView ArticleGoogle Scholar
  7. Hsu PL, Robbins H: Complete convergence and the law of large numbers. Proceedings of the National Academy of Sciences of the United States of America 1947, 33: 25–31. 10.1073/pnas.33.2.25MATHMathSciNetView ArticleGoogle Scholar
  8. Davis JA: Convergence rates for probabilities of moderate deviations. Annals of Mathematical Statistics 1968, 39: 2016–2028. 10.1214/aoms/1177698029MATHMathSciNetView ArticleGoogle Scholar
  9. Gut A, Spătaru A: Precise asymptotics in the law of the iterated logarithm. The Annals of Probability 2000,28(4):1870–1883. 10.1214/aop/1019160511MATHMathSciNetView ArticleGoogle Scholar
  10. Liu W, Lin Z: Precise asymptotics for a new kind of complete moment convergence. Statistics & Probability Letters 2006,76(16):1787–1799. 10.1016/j.spl.2006.04.027MATHMathSciNetView ArticleGoogle Scholar
  11. Kim T-S, Baek J-I: A central limit theorem for stationary linear processes generated by linearly positively quadrant-dependent process. Statistics & Probability Letters 2001,51(3):299–305. 10.1016/S0167-7152(00)00168-1MATHMathSciNetView ArticleGoogle Scholar
  12. Shao QM: A moment inequality and its applications. Acta Mathematica Sinica 1988,31(6):736–747.MATHMathSciNetGoogle Scholar

Copyright

© Jie Li. 2011

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.