Large Deviations for Sums of Independent Heavy-Tailed Random Variables

We obtain precise large deviations for heavy-tailed random sums , of independent random variables. are nonnegative integer-valued random variables independent of r.v. (X_i)i N 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.