© Xing-Cai Zhou et al. 2011
Received: 26 October 2010
Accepted: 27 January 2011
Published: 22 February 2011
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  has established the central limit theorem, Byrc and Smoleński  and Yang  have obtained moment inequalities and the strong law of large numbers, Wu [4, 5], Peligrad and Gut , and Gan  have studied almost sure convergence, Utev and Peligrad  have established maximal inequalities and the invariance principle, An and Yuan  have considered the complete convergence and Marcinkiewicz-Zygmund-type strong law of large numbers, and Budsaba et al.  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  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  and Erdös  proved the case and of the above theorem. The case and of the above theorem was proved by Spitzer . An and Yuan  studied the weighted sums of identically distributed -mixing sequence and have the following results.
Recently, Sung  obtained the following complete convergence results for weighted sums of identically distributed NA random variables.
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.
The proof of Theorem D was based on Theorem 1 of Chen et al. , which gave sufficient conditions about complete convergence for NA random variables. So far, it is not known whether the result of Chen et al.  holds for -mixing sequence. Hence, we use different methods from those of Sung . 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.
2. Proof of the Main Result
To obtain our results, the following lemmas are needed.
Lemma 2.1 (Utev and Peligrad ).
Hence, (1.7) is satisfied. From the proof of (2.1) of Sung , we obtain easily that the result holds.
Proof of Theorem 1.2.
Proof of Theorem 1.4.
The authors thank the Academic Editor and the reviewers for comments that greatly improved the paper. This work is partially supported by Anhui Provincial Natural Science Foundation (no. 11040606M04), Major Programs Foundation of Ministry of Education of China (no. 309017), National Important Special Project on Science and Technology (2008ZX10005-013), and National Natural Science Foundation of China (11001052, 10971097, and 10871001).
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