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 {Xn; 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}, {Xn; n ≥ 1} and {M(n); n ≥ 0} are mutually independent. Write S(n) = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum\limits_{i = 1}^{N(n)} {X_i - cM(n)} $$\end{document}. Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.