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Báo cáo hóa học: "Research Article Some Strong Limit Theorems for Weighted Product Sums of ρ-Mixing Sequences of Random Variables"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Some Strong Limit Theorems for Weighted Product Sums of ρ-Mixing Sequences of Random Variables | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 174768 10 pages doi 10.1155 2009 174768 Research Article Some Strong Limit Theorems for Weighted Product Sums of p-Mixing Sequences of Random Variables Qunying Wu and Yuanying Jiang College of Science Guilin University of Technology Guilin 541004 China Correspondence should be addressed to Qunying Wu wqy666@glite.edu.cn Received 4 June 2009 Revised 28 September 2009 Accepted 18 November 2009 Recommended by Jewgeni Dshalalow We study almost sure convergence for p-mixing sequences of random variables. Many of the previous results are our special cases. For example the authors extend and improve the corresponding results of Chen et al. 1996 and Wu and Jiang 2008 . We extend the classical Jamison convergence theorem and the Marcinkiewicz strong law of large numbers for independent sequences of random variables to p-mixing sequences of random variables without necessarily adding any extra conditions. Copyright 2009 Q. Wu and Y. Jiang. 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. 1. Introduction and Lemmas Let Q F P be a probability space. The random variables we deal with are all defined on Q F P . Let Xn n 1 be a sequence of random variables. For each nonempty set S c N and write FS ơ Xi i e S . Given ơ-algebras B R in F let p B R sup corr X Y X e L2 B Y e L2 R 1.1 where corr X Y EXY - EXEY YVar X Var Y. Define the p-mixing coefficients by p n sup p FS Ft finite subsets S T c N such that dist S T n n 0. 1.2 Obviously 0 p n 1 p n 1 n 0 and p 0 1 except in the trivial case where all of the random variables Xi are degenerate. Definition 1.1. A random variables sequence Xn n 1 is said to be a p-mixing random variables sequence if there exists k e N such that p k 1. 2 Journal of Inequalities and Applications p-mixing