International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 3, Pages 481-497
doi:10.1155/S0161171292000644

Some results on convergence rates for probabilities of moderate deviations for sums of random variables

Deli Li,1 Xiangchen Wang,2 and M. Bhaskara Rao3

1Institute of Mathematics, Jilin University, Changchun 130023, China
2Department of Mathematics, Jilin University, Changchun 130023, China
3Department of Statistics, North Dakota State University, Fargo 58105, ND, USA

Received 10 April 1991; Revised 1 January 1992

Copyright © 1992 Deli Li et al. 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.

Abstract

Let X, Xn, n≥1 be a sequence of iid real random variables, and Sn=∑k=1nXk, n≥1. Convergence rates of moderate deviations are derived, i.e., the rate of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of series ∑n≥1(ψ2(n)/n)P(|Sn|≥nφ(n)) only under the assumptions convergence that EX=0 and EX2=1, where φ and ψ are taken from a broad class of functions. These results generalize and improve some recent results of Li (1991) and Gafurov (1982) and some previous work of Davis (1968). For b∈[0,1] and ϵ>0, letλϵ,b=∑n≥3((loglogn)b/n)I(|Sn|≥(2+ϵ)nloglogn).The behaviour of Eλϵ,b as ϵ↓0 is also studied.