Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 972324, 9 pages
Research Article

Second Moment Convergence Rates for Uniform Empirical Processes

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received 21 May 2010; Revised 3 August 2010; Accepted 19 August 2010

Academic Editor: Andrei Volodin

Copyright © 2010 You-You Chen and Li-Xin 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 αn(t)=n1/2i=1n(I{Uit}t),0t1, αn=sup0t1|αn(t)|. In this paper, we get the exact convergence rates of weighted infinite series of Eαn2I{αnε(logn)1/β}.