Large deviation results for generalized compound negative binomial risk models

In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {X _n ; n ≥ 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈ R ⁺, {N(n); n ≥ 0} be a sequence of negative binomial distributed random variables with a parameter p ∈ (0, 1), n ≥ 0, let {M(n); n ≥ 0} be a Poisson process with intensity λ > 0. Suppose {N(n); n ≥ 0}, {X _n ; n ≥ 1} and {M(n); n ≥ 0} are mutually independent. Write S(n) = . Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.