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

- Jie Li
^{1, 2}Email author

**2011**:320932

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

© Jie Li. 2011

**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

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

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

Theorem A.

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

Theorem B.

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.

Theorem 1.2.

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.

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

Lemma 1.5 (see [2]).

Lemma 1.6 (see [12]).

## 2. Proofs

Proof of Theorem 1.1.

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

Proposition 2.1.

Proof.

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

Proposition 2.2.

Proof.

(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 [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

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