Large deviations for sums of independent heavy-tailed random variables

We obtain precise large deviations for heavy-tailed random sums S(t) = ∑ i = 1 N (t) X i , t ≥ 0, of independent random variables. (N (t)) t ≥ 0 (are nonnegative integer-valued random variables independent of r.v. (X) ii εN with distribution functions F i . We assume that the average of right tails of distribution functions F i is equivalent to some distribution function with regularly varying tail. An example with the Pareto law as the limit function is given.