期刊名称：International Journal of Mathematics and Mathematical Sciences
印刷版ISSN：0161-1712
电子版ISSN：1687-0425
出版年度：2004
卷号：2004
DOI：10.1155/S0161171204406516
出版社：Hindawi Publishing Corporation
摘要：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.
