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

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,

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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

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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

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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

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where .

Theorem 1.2.

Under the conditions in Theorem 1.1, one has

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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

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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

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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

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Proof of Theorem 1.1.

Without loss of generality, we assume that . We have

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

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By Proposition 5.1 in [10], we have

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

Proposition 2.1.

One has

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Proof.

Write

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where

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Since implies , we have

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For , by Markov's inequality, we get

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

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

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Set . As , by (2.10), we have

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So, when ,

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By (2.12), we have

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

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Then,

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So, we get

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Therefore,

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By Lemma 1.6, noting that , for ,

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For , we have

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Then, for , , we have

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For , we decompose it into two parts,

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It is easy to see that

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So,

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Now, we estimate , by (2.23),

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For , we have

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

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Finally, , and we will get

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then

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

Proposition 2.2.

One has

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Proof.

Consider the following:

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

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Hence,

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Now, we estimate . Here, , so

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We have

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

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then

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By Lemma 1.6, for , we have

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For , we have

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Next, turning to , it follows that

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then

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For , it follows that

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Finally, , we have

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

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(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

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So, we only prove the following two propositions:

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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.