Large Deviations for Sums of Independent Heavy-Tailed Random Variables

Abstract

We obtain precise large deviations for heavy-tailed random sums $$S(t) = \sum\nolimits_{i = 1}^{N(t)} {X_i ,t \geqslant 0} $$ , of independent random variables. $$(N(t))_{t \geqslant 0} $$ are nonnegative integer-valued random variables independent of r.v. (X i )i $$ \in $$ 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.