© Xing-cai Zhou and Jin-guan Lin. 2010
Received: 15 May 2010
Accepted: 21 October 2010
Published: 25 October 2010
The -mixing random variables were first introduced by Kolmogorov and Rozanov . The limiting behavior of -mixing random variables is very rich, for example, these in the study by Ibragimov , Peligrad , and Bradley  for central limit theorem; Peligrad  and Shao [6, 7] for weak invariance principle; Shao  for complete convergence; Shao  for almost sure invariance principle; Peligrad , Shao  and Liang and Yang  for convergence rate; Shao , for the maximal inequality, and so forth.
For arrays of rowwise independent random variables, complete convergence has been extensively investigated (see, e.g., Hu et al. , Sung et al. , and Kruglov et al. ). Recently, complete convergence for arrays of rowwise dependent random variables has been considered. We refer to Kuczmaszewska  for -mixing and -mixing sequences, Kuczmaszewska  for negatively associated sequence, and Baek and Park  for negatively dependent sequence. In the paper, we study the complete convergence for arrays of rowwise -mixing sequence under some suitable conditions using the techniques of Kuczmaszewska [16, 17]. We consider the case of complete convergence of maximum weighted sums, which is different from Kuczmaszewska . Some results also generalize some previous known results for rowwise independent random variables.
Now, we present a few definitions needed in the coming part of this paper.
Throughout the sequel, will represent a positive constant although its value may change from one appearance to the next; indicates the maximum integer not larger than ; denotes the indicator function of the set .
The following lemmas will be useful in our study.
Lemma 1.4 (Shao ).
Lemma 1.5 (Sung ).
Lemma 1.6 (Zhou ).
This paper is organized as follows. In Section 2, we give the main result and its proof. A few applications of the main result are provided in Section 3.
2. Main Result and Its Proof
Now, we state our main result.
Let be an array of rowwise -mixing random variables satisfying for some , and let be an array of real numbers. Let be an increasing sequence of positive integers, and let be a sequence of positive real numbers. If for some and any the following conditions are fulfilled:
Theorem 2.1 firstly gives the condition of the mixing coefficient, so the conditions (a)–(c) do not contain the mixing coefficient. Thus, the conditions (a)–(c) are obviously simpler than the conditions (i)–(iii) in Theorem 2.1 of Kuczmaszewska . Our conditions are also different from those of Theorem 2.1 in the study by Kuczmaszewska : is only required in Theorem 2.1, not in Theorem 2.1 of Kuczmaszewska ; the powers of in (b) and (c) of Theorem 2.1 are and , respectively, not in Theorem 2.1 of Kuczmaszewska .
Now, we give the proof of Theorem 2.1.
From (b), (c), and (2.5), we see that (2.4) holds.
In order to prove that (c) holds, we consider the following two cases.
Let be an array of rowwise -mixing random variables satisfying for some , and let be an array of real numbers. Let be a slowly varying function as . If for some and real number , and any the following conditions are fulfilled:
which proves that condition (A) is satisfied.
which proves that (B) holds.
In order to prove that (C) holds, we consider the following two cases.
We complete the proof of the theorem.
The authors thank the academic editor and the reviewers for comments that greatly improved the paper. This work is partially supported by the Anhui Province College Excellent Young Talents Fund Project of China (no. 2009SQRZ176ZD) and National Natural Science Foundation of China (nos. 11001052, 10871001, 10971097).
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