Precise Asymptotics in the Law of Iterated Logarithm for Moving Average Process under Dependence
© Jie Li. 2011
Received: 10 November 2010
Accepted: 3 March 2011
Published: 15 March 2011
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
where . Many limiting results of moving average for -mixing have been obtained. For example, Zhang  got complete convergence.
Li and Zhang  achieved precise asymptotics in the law of the iterated logarithm.
On the other hand, since Hsu and Robbins  introduced the concept of the complete convergence, there have been extensions in some directions. For the case of i.i.d. random variables, Davis  proved , for if and only if . Gut and Spătaru  gave the precise asymptotics of . We know that complete convergence can be derived from complete moment convergence. Liu and Lin  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.
We first proceed with some useful lemmas.
The proof is similar to Theorem 1 in . Set . From Lemma 1.4, one can get as .
Lemma 1.5 (see ).
Lemma 1.6 (see ).
Proof of Theorem 1.1.
Hence, Theorem 1.1 will be proved if we show the following two propositions.
Hence, (2.4) can be referred from (2.9), (2.17), (2.20), (2.26), and (2.28).
(2.29) can be derived by (2.32), (2.36), and (2.43).
Proof of Theorem 1.2.
The proof of (2.45) can be referred to , and the proof of (2.46) is similar to Propositions 2.1 and 2.2.
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 )
- Ibragimov IA: Some limit theorems for stationary processes. Theory of Probability and Its Applications 1962, 7: 349–382. 10.1137/1107036View ArticleGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- Davis JA: Convergence rates for probabilities of moderate deviations. Annals of Mathematical Statistics 1968, 39: 2016–2028. 10.1214/aoms/1177698029MATHMathSciNetView ArticleGoogle Scholar
- 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
- 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
- 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
- Shao QM: A moment inequality and its applications. Acta Mathematica Sinica 1988,31(6):736–747.MATHMathSciNetGoogle Scholar
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.