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

*Journal of Inequalities and Applications*
**volumeÂ 2011**, ArticleÂ number:Â 320932 (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

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,

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

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

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 SpÄƒtaru [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

where .

Theorem 1.2.

Under the conditions in Theorem 1.1, one has

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

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

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

## 2. Proofs

Proof of Theorem 1.1.

Without loss of generality, we assume that . We have

Set , where . By Theorem B, we need to show

By Proposition 5.1 in [10], we have

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

Proposition 2.1.

One has

Proof.

Write

where

Since implies , we have

For , by Markov's inequality, we get

From (2.7) and (2.8), we can get

Note that , where . By Lemma 1.5, we can assume that

Set . As , by (2.10), we have

So, when ,

By (2.12), we have

Set , then (referred by [4]). We can get

Then,

So, we get

Therefore,

By Lemma 1.6, noting that , for ,

For , we have

Then, for , , we have

For , we decompose it into two parts,

It is easy to see that

So,

Now, we estimate , by (2.23),

For , we have

From (2.24) and (2.25), we can get

Finally, , and we will get

then

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

Proposition 2.2.

One has

Proof.

Consider the following:

We first estimate , for , by Markov's inequality,

Hence,

Now, we estimate . Here, , so

We have

We estimate first. Similar to the proof of (2.16), we have

then

By Lemma 1.6, for , we have

For , we have

Next, turning to , it follows that

then

For , it follows that

Finally, , we have

From (2.38) to (2.42), we can get

(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

So, we only prove the following two propositions:

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.

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## 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 )

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Li, J. Precise Asymptotics in the Law of Iterated Logarithm for Moving Average Process under Dependence.
*J Inequal Appl* **2011**, 320932 (2011). https://doi.org/10.1155/2011/320932

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