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

Let {Xn;n[greater-or-equal, slanted]1} be an arbitrary sequence of heavy-tailed random variables, independent of a process of nonnegative, integer-valued rv's {N(t);t[greater-or-equal, slanted]0}. In this paper, we present a necessary and a sufficient condition on {N(t);t[greater-or-equal, slanted]0} under which the precise large deviation result for nonrandom sum can be extended to the random sum SN(t). The results we obtain not only generalize but also improve some related classical ones.