1. Introduction

Suppose that is a sequence of random variables and is a subset of the natural number set . Let ,

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where

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Definition 1.1.

A random variable sequence is said to be a -mixing random variable sequence if there exists such that .

The notion of -mixing seems to be similar to the notion of -mixing, but they are quite different from each other. Many useful results have been obtained for -mixing random variables. For example, Bradley [1] has established the central limit theorem, Byrc and Smoleński [2] and Yang [3] have obtained moment inequalities and the strong law of large numbers, Wu [4, 5], Peligrad and Gut [6], and Gan [7] have studied almost sure convergence, Utev and Peligrad [8] have established maximal inequalities and the invariance principle, An and Yuan [9] have considered the complete convergence and Marcinkiewicz-Zygmund-type strong law of large numbers, and Budsaba et al. [10] have proved the rate of convergence and strong law of large numbers for partial sums of moving average processes based on -mixing random variables under some moment conditions.

For a sequence of . random variables, Baum and Katz [11] proved the following well-known complete convergence theorem: suppose that is a sequence of . random variables. Then and if and only if for all .

Hsu and Robbins [12] and Erdös [13] proved the case and of the above theorem. The case and of the above theorem was proved by Spitzer [14]. An and Yuan [9] studied the weighted sums of identically distributed -mixing sequence and have the following results.

Theorem B.

Let be a -mixing sequence of identically distributed random variables, , , and suppose that for . Assume that is an array of real numbers satisfying

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If , then

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Theorem C.

Let be a -mixing sequence of identically distributed random variables, , , and for . Assume that is array of real numbers satisfying (1.3). Then

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Recently, Sung [15] obtained the following complete convergence results for weighted sums of identically distributed NA random variables.

Theorem D.

Let be a sequence of identically distributed NA random variables, and let be an array of constants satisfying

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for some . Let for some . Furthermore, suppose that where . If

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then

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We find that the proof of Theorem C is mistakenly based on the fact that (1.5) holds for . Hence, the Marcinkiewicz-Zygmund-type strong laws for -mixing sequence have not been established.

In this paper, we shall not only partially generalize Theorem D to -mixing case, but also extend Theorem B to the case . The main purpose is to establish the Marcinkiewicz-Zygmund strong laws for linear statistics of -mixing random variables under some suitable conditions.

We have the following results.

Theorem 1.2.

Let be a sequence of identically distributed -mixing random variables, and let be an array of constants satisfying

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where for some and . Let . If for and (1.8) for , then (1.9) holds.

Remark 1.3.

The proof of Theorem D was based on Theorem 1 of Chen et al. [16], which gave sufficient conditions about complete convergence for NA random variables. So far, it is not known whether the result of Chen et al. [16] holds for -mixing sequence. Hence, we use different methods from those of Sung [15]. We only extend the case of Theorem D to -mixing random variables. It is still open question whether the result of Theorem D about the case holds for -mixing sequence.

Theorem 1.4.

Under the conditions of Theorem 1.2, the assumptions for and (1.8) for imply the following Marcinkiewicz-Zygmund strong law:

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2. Proof of the Main Result

Throughout this paper, the symbol represents a positive constant though its value may change from one appearance to next. It proves convenient to define , where denotes the natural logarithm.

To obtain our results, the following lemmas are needed.

Lemma 2.1 (Utev and Peligrad [8]).

Suppose is a positive integer, , and . Then there exists a positive constant such that the following statement holds.

If is a sequence of random variables such that with and for every , then for all ,

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where .

Lemma 2.2.

Let be a random variable and be an array of constants satisfying (1.10), . Then

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Proof.

If , by and Lyapounov's inequality, then

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Hence, (1.7) is satisfied. From the proof of (2.1) of Sung [15], we obtain easily that the result holds.

Proof of Theorem 1.2.

Let . For all , we have

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To obtain (1.9), we need only to prove that and .

By Lemma 2.2, one gets

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Before the proof of , we prove firstly

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For ,

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For ,

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Thus (2.6) holds. So, to prove , it is enough to show that

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By the Chebyshev inequality and Lemma 2.1, for , we have

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For , we consider the following two cases.

If , note that . We have

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If , note that . we have

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Next, we prove in the following two cases.

If or , take . Noting that , we have

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If or , one gets . Since , it implies . Therefore, we have

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for all . Hence, . Taking , we have

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Proof of Theorem 1.4.

By (1.9), a standard computation (see page 120 of Baum and Katz [11] or page 1472 of An and Yuan [9]), and the Borel-Cantelli Lemma, we have

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For any , there exists an integer such that . So

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From (2.16) and (2.17), we have

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