A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums S( t) = Xi, t≥0, where {Xn, 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 { Xn, 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.