International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 66, Pages 3565-3576
doi:10.1155/S0161171204406516

Precise lim sup behavior of probabilities of large deviations for sums of i.i.d. random variables

Deli Li1 and Andrew Rosalsky2

1Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada
2Department of Statistics, University of Florida, Gainesville 32611-8545, FL, USA

Received 23 June 2004

Copyright © 2004 Hindawi Publishing Corporation. 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 real-valued i.i.d. random variables and let Sn=∑i=1nXi, n≥1. In this paper, we study the probabilities of large deviations of the form P(Sn>tn1/p), P(Sn<−tn1/p), and P(|Sn|>tn1/p), where t>0 and 0<p<2. We obtain precise asymptotic estimates for these probabilities under mild and easily verifiable conditions. For example, we show that if Sn/n1/p→P0 and if there exists a nonincreasing positive function ϕ(x) on [0,∞) which is regularly varying with index α≤−1 such that limsupx→∞P(|X|>x1/p)/ϕ(x)=1, then for every t>0, limsupn→∞P(|Sn|>tn1/p)/(nϕ(n))=tpα.