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Báo cáo hóa học: " Research Article Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing 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 Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 576301 9 pages doi 10.1155 2011 576301 Research Article Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables Daxiang Ye and Qunying Wu College of Science Guilin University of Technology Guilin 541004 China Correspondence should be addressed to Daxiang Ye 3040801111@163.com Received 19 September 2010 Revised 1 January 2011 Accepted 26 January 2011 Academic Editor Ondrej Dosly Copyright 2011 D. Ye and Q. Wu. 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 here an almost sure central limit theorem for product of sums of strongly mixing positive random variables. 1. Introduction and Results In recent decades there has been a lot of work on the almost sure central limit theorem ASCLT we can refer to Brosamler 1 Schatte 2 Lacey and Philipp 3 and Peligrad and Shao 4 . Khurelbaatar and Rempala 5 gave an ASCLT for product of partial sums of i.i.d. random variables as follows. Theorem 1.1. Let Xn n 1 be a sequence ofi.i.d. positive random variables with EXi n 0 and Var Xi Ơ2. Denote Y ơ ụ. the coefficient of variation. Then for any real x lim T1- y 11 í ịẠ x F x a.s 1.1 n i ln nkffi k k ffi where Sn 2n 1 Xk 1 is the indicator function F - is the distribution function of the random variable eN and N is a standard normal variable. Recently Jin 6 had proved that 1.1 holds under appropriate conditions for strongly mixing positive random variables and gave an ASCLT for product of partial sums of strongly mixing as follows. 2 Journal of Inequalities and Applications Theorem 1.2. Let Xn n 1 be a sequence of identically distributed positive strongly mixing random variable with EX1 ỳ 0 and Var X1 ơ2 dk 1 k Dn n 1 dk. Denote by Y ơ ỳ the coefficient of variation ơn Var n 1 Sk - .