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Báo cáo hóa học: "Research Article An Exponential Inequality for Negatively Associated 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 An Exponential Inequality for Negatively Associated Random Variables | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 649427 7 pages doi 10.1155 2009 649427 Research Article An Exponential Inequality for Negatively Associated Random Variables 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 15 October 2008 Revised 16 February 2009 Accepted 7 May 2009 Recommended by Jewgeni Dshalalow An exponential inequality is established for identically distributed negatively associated random variables which have the finite Laplace transforms. The inequality improves the results of Kim and Kim 2007 Nooghabi and Azarnoosh 2009 and Xing et al. 2009 . We also obtain the convergence rate O 1 n1 2 logn -1 2 for the strong law of large numbers which improves the corresponding ones of Kim and Kim Nooghabi and Azarnoosh and Xing et al. Copyright 2009 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 Let Xn n 1 be a sequence of random variables defined on a fixed probability space Q F P The concept of negatively associated random variables was introduced by Alam and Saxena 1 and carefully studied by Joag-Dev and Proschan 2 . A finite family of random variables Xi 1 i n is said to be negatively associated if for every pair of disjoint subsets A and B of 1 2 . n Cov f1 Xi i e AfXj j e BỴ 0 1.1 whenever f1 and f2 are coordinatewise increasing and the covariance exists. An infinite family of random variables is negatively associated if every finite subfamily is negatively associated. As pointed out and proved by Joag-Dev and Proschan 2 a number of well-known multivariate distributions possess the negative association property such as multinomial convolution of unlike multinomial multivariate .