On Complete Convergence for Arrays of Rowwise -Mixing Random Variables and Its Applications
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Research Article On Complete Convergence for Arrays of Rowwise ρ-Mixing Random Variables and Its Applications Xing-cai Zhou1, 2 and Jin-guan Lin1 1 2
Department of Mathematics, Southeast University, Nanjing 210096, China Department of Mathematics and Computer Science, Tongling University, Tongling, Anhui 244000, China
Correspondence should be addressed to Jin-guan Lin, [email protected] Received 15 May 2010; Revised 23 August 2010; Accepted 21 October 2010 Academic Editor: Soo Hak Sung Copyright q 2010 X.-c. Zhou and J.-g. Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give out a general method to prove the complete convergence for arrays of rowwise ρ-mixing random variables and to present some results on complete convergence under some suitable conditions. Some results generalize previous known results for rowwise independent random variables.
1. Introduction Let {Ω, F, P } be a probability space, and let {Xn ; n ≥ 1} be a sequence of random variables defined on this space. Definition 1.1. The sequence {Xn ; n ≥ 1} is said to be ρ-mixing if
ρn sup k≥1
⎧ ⎪ ⎨
⎫ ⎪ ⎬
|EXY − EXEY | −→ 0 sup ⎪ ⎪ ∞ ⎩ X∈L2 F1k , Y ∈L2 Fnk EX − EX2 EY − EY 2 ⎭
1.1
n as n → ∞, where Fm denotes the σ-field generated by {Xi ; m ≤ i ≤ n}.
The ρ-mixing random variables were first introduced by Kolmogorov and Rozanov 1. The limiting behavior of ρ-mixing random variables is very rich, for example, these in the study by Ibragimov 2, Peligrad 3, and Bradley 4 for central limit theorem; Peligrad 5 and Shao 6, 7 for weak invariance principle; Shao 8 for complete convergence; Shao
2
Journal of Inequalities and Applications
9 for almost sure invariance principle; Peligrad 10, Shao 11 and Liang and Yang 12 for convergence rate; Shao 11, 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. 13, Sung et al. 14, and Kruglov et al. 15. Recently, complete convergence for arrays of rowwise dependent random variables has been considered. We refer to Kuczmaszewska 16 for ρ-mixing and ρ-mixing
sequences, Kuczmaszewska 17 for negatively associated sequence, and Baek and Park 18 for negatively dependent sequence. In the paper, we study the complete convergence for arrays of rowwise ρ-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 16. 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. Definition 1.2. An array {Xni ; i ≥ 1, n ≥ 1} of random variables is said to be stochastically dominated by a random variable X if there exists a const
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