International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 253750, 15 pages
Research Article

Convergence Rates for Probabilities of Moderate Deviations for Multidimensionally Indexed Random Variables

Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2, Canada

Received 12 March 2009; Revised 23 August 2009; Accepted 27 September 2009

Academic Editor: Jewgeni Dshalalow

Copyright © 2009 Dianliang Deng. 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 {X,Xn¯;n¯Z+d} be a sequence of i.i.d. real-valued random variables, and Sn¯=k¯n¯Xk¯, n¯Z+d. Convergence rates of moderate deviations are derived; that is, the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of the series n¯b(n¯)ψ2(a(n¯))P{|Sn¯|a(n¯)ϕ(a(n¯))}, where a(n¯)=n11/α1nd1/αd, b(n¯)=n1β1ndβd, ϕ and ψ are taken from a broad class of functions. These results generalize and improve some results of Li et al. (1992) and some previous work of Gut (1980).