Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article On Inverse Moments for a Class of Nonnegative Random Variables | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 823767 13 pages doi 2010 823767 Research Article On Inverse Moments for a Class of Nonnegative Random Variables Soo Hak Sung Department of Applied Mathematics Pai Chai University Taejon 302-735 Republic of Korea Correspondence should be addressed to Soo Hak Sung sungsh@ Received 1 April 2010 Accepted 20 May 2010 Academic Editor Andrei Volodin Copyright 2010 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. Using exponential inequalities Wu et al. 2009 and Wang et al. 2010 obtained asymptotic approximations of inverse moments for nonnegative independent random variables and nonnegative negatively orthant dependent random variables respectively. In this paper we improve and extend their results to nonnegative random variables satisfying a Rosenthal-type inequality. 1. Introduction Let Zn n 1 be a sequence of nonnegative random variables with finite second moments. Let us denote Xn n Ẻ Var Zj . n M We will establish that under suitable conditions the inverse moment can be approximated by the inverse of the moment. More precisely we will prove that E a Xn a a EXnỴa where a 0 a 0 and cn dn means that cndn1 1 as n TO. The left-hand side of is the inverse moment and the right-hand side is the inverse of the moment. Generally it is not easy to compute the inverse moment but it is much easier to compute the inverse of the moment. 2 Journal of Inequalities and Applications The inverse moments can be applied in many practical applications. For example they appear in Stein estimation and Bayesian poststratification see Wooff 1 and Pittenger 2 evaluating risks of estimators and powers of test statistics see Marciniak and Wesotowski 3 and Fujioka 4 expected relaxation times of complex
