International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 66, Pages 3565-3576
Precise lim sup behavior of probabilities of large deviations for
sums of i.i.d. random variables
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 Deli Li and Andrew Rosalsky. 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 be a sequence of real-valued i.i.d. random variables and
let , . In this paper, we study the probabilities of large deviations of
the form , , and , where and . We obtain precise asymptotic estimates for these
probabilities under mild and easily
verifiable conditions. For example, we show that if
and if there exists a nonincreasing positive
which is regularly varying with index
, then for every , .