tailieunhanh - Báo cáo hóa học: " Research Article Moment Inequalities and Complete Moment Convergence"

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 Moment Inequalities and Complete Moment Convergence | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 271265 14 pages doi 2009 271265 Research Article Moment Inequalities and Complete Moment Convergence Soo Hak Sung Department of Applied Mathematics Pai Chai University Taejon 302-735 South Korea Correspondence should be addressed to Soo Hak Sung sungsh@ Received 22 August 2009 Accepted 26 September 2009 Recommended by Andrei Volodin Let Yi 1 i n and Zi 1 i n be sequences of random variables. For any e 0 and a 0 bounds for E 2n 1 Yi Zi - ea and E max1 k n E1fYi Zi - ea are obtained. From these results we establish general methods for obtaining the complete moment convergence. The results of Chow 1988 Zhu 2007 and Wu and Zhu 2009 are generalized and extended from independent or dependent random variables to random variables satisfying some mild conditions. Some applications to dependent random variables are discussed. 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 most interesting inequalities to probability theory are probably Marcinkiewicz-Zygmund and Rosenthal inequalities. For a sequence Xi 1 i n of . random variables with E X1 q TO for some q 1 Marcinkiewicz and Zygmund 1 and Rosenthal 2 1 q 2 and q 2 resp. proved that there exist positive constants Aq and Bq depending only on q such that E Ẻ Xi - EXi Aq E Xi iq i 1 i 1 for 1 q 2 n E Xi - EXi i 1 i 1 n ZE Xi 2 i 1 for q 2. E q bJ p X q n 2 Journal of Inequalities and Applications The following Marcinkiewicz-Zygmund and Rosenthal type maximal inequalities are well known. For a sequence Xi 1 i n of . random variables with E X1 q TO for some q 1 there exist positive constants Cq and Dq .