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: HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES | HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES ISHA DEWAN AND B. L. S. PRAKASA RAO Received 21 April 2005 Revised 26 October 2005 Accepted 11 December 2005 Let Yn n 1 be a sequence of nonmonotonic functions of associated random variables. We derive a Newman and Wright 1981 type of inequality for the maximum of partial sums of the sequence Yn n 1 and a Hajek-Renyi-type inequality for nonmonotonic functions of associated random variables under some conditions. As an application a strong law of large numbers is obtained for nonmonotonic functions of associated random varaibles. Copyright 2006 I. Dewan and B. L. S. P. Rao. 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 O ty a probability space and Xn n 1 be a sequence of associated random variables defined on it. A finite collection X1 X2 . Xn is said to be associated if for every pair of functions h x and g x from Rn to R which are nondecreasing componentwise Cov h X g X 0 whenever it is finite where X X1 X2 . Xn . The infinite sequence Xn n 1 is said to be associated if every finite subfamily is associated. Associated random variables are of considerable interest in reliability studies cf. Barlow and Proschan 1 Esary et al. 6 statistical physics cf. Newman 9 10 and percolation theory cf. Cox and Grimmet 4 . For an extensive review of several probabilistic and statistical results for associated sequences see Roussas 14 and Dewan and Rao 5 . Newman and Wright 12 proved an inequality for maximum of partial sums and Prakasa Rao 13 proved the Hajek-Renyi-type inequality for associated random variables. Esary et al. 6 proved that monotonic functions of associated random variables are associated. Hence one can easily extend the above-mentioned inequalities to monotonic Hindawi Publishing .
