A contribution to large deviations for heavy-tailed random sums

Abstract

In this paper we consider the large deviations for random sums S ( t )=Σ i =1 N(t) X i , t ≥0, where { X n , n ≥1｝ are independent, identically distributed and non _ negative random variables with a common heavy-tailed distribution function F , and ｛ N ( t ), t ≥0｝ is a process of non-negative integer-valued random variables, independent of { X n , n ≥1｝. Under the assumption that the tail of F is of Pareto's type (regularly or extended regularly varying), we investigate what reasonable condition can be given on ｛ N ( t ), t ≥0｝ under which precise large deviation for S ( t ) holds. In particular, the condition we obtain is satisfied for renewal counting processes.