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

Let {Xn;n⩾1}{Xn;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}{N(t);t⩾0}. In this paper, we present a necessary and a sufficient condition on {N(t);t⩾0}{N(t);t⩾0} under which the precise large deviation result for nonrandom sum Sn=∑i=1nXi can be extended to the random sum SN(t)SN(t). The results we obtain not only generalize but also improve some related classical ones.