A contribution to large deviations for heavy-tailed random sums

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