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

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.