The general principle for precise large deviations of heavy-tailed random sums

Let{X-n;n >= 1) be an arbitrary sequence of heavy-tailed random variables, independent of a process of nonnegative, integer-valued rv's {N(t);t >= 0}. In this paper, we present a necessary and a sufficient condition on {N(t);t >= 0} under which the precise large deviation result for nonrandorn sum S-n = Sigma(n) (i=1) X-i can be extended to the random sum S-N(t). The results we obtain not only generalize but also improve some related classical ones. (c) 2007 Elsevier B. V. All rights reserved.