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Complete convergence and almost sure convergence of weighted sums of random variables
Authors:Deli Li  M Bhaskara Rao  Tiefeng Jiang  Xiangchen Wang
Institution:(1) Institute of Mathematics, Jilin University, 130023 Changchun, China;(2) Department of Mathematics, University of Regina, SK S4S 0A2, Canada;(3) Department of Statistics, North Dakota State University, 58105 Fargo, North Dakota;(4) Department of Mathematics, Jilin University, 130023 Changchun, China
Abstract:Letr>1. For eachnge1, let {X nk , –infin<k<infin} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee 
$$\Sigma _{n \geqslant 1} n^{r - 2} P\{ |\Sigma _{k =  - \infty }^\infty  X_{nk} |  \geqslant \varepsilon \}< \infty $$
for every epsi>0. This result is used to establish some results on complete convergence for weighted sums of independent random variables. The main idea is that we devise an effetive way of combining a certain maximal inequality of Hoffmann-Jørgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent random variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent random variables. Convergence rates in the almost sure convergence of some summability methods ofiid random variables are also established.
Keywords:Almost sure convergence  complete convergence  comparison principle  Hoffmann-Jø  rgensen's inequality  summability methods  symmetrization  weighted sums
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