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Báo cáo hóa học: " Research Article Second Moment Convergence Rates for Uniform Empirical Processes"

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 Second Moment Convergence Rates for Uniform Empirical Processes | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 972324 9 pages doi 10.1155 2010 972324 Research Article Second Moment Convergence Rates for Uniform Empirical Processes You-You Chen and Li-Xin Zhang Department of Mathematics Zhejiang University Hangzhou 310027 China Correspondence should be addressed to You-You Chen cyyooo@gmail.com Received 21 May 2010 Revised 3 August 2010 Accepted 19 August 2010 Academic Editor Andrei Volodin Copyright 2010 Y.-Y. Chen and L.-X. Zhang. 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. Let U1 U2 . Un be a sequence of independent and identically distributed U 0 1 -distributed random variables. Define the uniform empirical process as an t in 1 2 Xt 1 I Ui t - t 0 t 1 anH sup0 t 1 an t . In this paper we get the exact convergence rates of weighted infinite series of E an 2I an e logn 1 . 1. Introduction and Main Results Let X Xn n 1 be a sequence of independent and identically distributed i.i.d. random variables with zero mean. Set Sn n 1 Xi for n 1 and logx ln x V e . Hsu and Robbins 1 introduced the concept of complete convergence. They showed that yP Sn en TO e 0 n 1 1.1 if EX 0 and EX2 TO. The converse part was proved by the study of Erdos in 2 . Obviously the sum in 1.1 tends to infinity as e 0. Many authors studied the exact rates in terms of e cf. 3-5 . Chow 6 studied the complete convergence of E Sn - ena e 0. Recently Liu and Lin 7 introduced a new kind of complete moment convergence which is interesting and got the precise rate of it as follows. 2 Journal of Inequalities and Applications Theorem A. Suppose that X Xn n 1 is a sequence ofi.i.d. random variables then i iỉ n1 Sn en 2 C1-2 e lo5 e n 1 n holds if and only if EX 0 EX2 Ơ2 and EX2log X TO. Other than partial sums many authors investigated precise rates in some .