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Báo cáo hóa học: " Research Article Maximal Inequalities for Dependent Random Variables and Applications"

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 Maximal Inequalities for Dependent Random Variables and Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 598319 10 pages doi 10.1155 2008 598319 Research Article Maximal Inequalities for Dependent Random Variables and Applications Soo Hak Sung Department of Applied Mathematics Pai Chai University Taejon 302-735 South Korea Correspondence should be addressed to Soo Hak Sung sungsh@mail.pcu.ac.kr Received 16 April 2008 Revised 3 June 2008 Accepted 7 July 2008 Recommended by Ondrej Dosly For a sequence Xn n 1 of dependent square integrable random variables and a sequence bn n 1 of positive numbers we establish a maximal inequality for weighted sums of dependent random variables. Applying this inequality we obtain the almost sure convergence of n 1Xi bi and Sn 1Xi bn. Copyright 2008 Soo Hak Sung. 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 Throughout this paper let Xn n 1 be a sequence of random variables defined on a probability space Q F P and let bn n 1 be a sequence of positive numbers. We assume that there exists a sequence pn n 1 of nonnegative constants such that supE XkXk n pn for n 1. 1.1 k 1 In this paper we establish a maximal inequality for weighted sums of the dependent random variables satisfying 1.1 . Applying this inequality we obtain under some suitable conditions on the sequence pn that n X converges a.s. as n t TO 1.2 bi i 1 L and the strong law of large numbers SLLN n 1Xi -t 0 a.s. 1.3 bn Note that if 0 bn T TO then 1.2 implies 1.3 by the Kronecker lemma. 2 Journal of Inequalities and Applications For a sequence of dependent random variables satisfying 1.1 the SLLNs were established by Hu et al. 1 2 and Lyons 3 . Lyons 3 obtained an SLLN under the conditions that Var Xn 0 1 and bn n. Without condition Var Xn 0 1 Hu et al. 1 obtained an SLLN where bn n. Hu et al. 2 also obtained .