The Asymptotic Behavior Of Precise Large Deviations Of Heavy-tailed Random Sums

In insurance, usually an insurance policy can be seen as a random variable, and them the important risk which the insurance companies face is a number of large claims. However, in the limit theory of large deviations theory, the main object of study is the asymptotic behavior, i.e. when xâ†'∞, the probability P(·>x), which is commonly called the ruin probability. Therefore, large deviations theory has been widely applied to the insurance and the financial industry.In this paper, we study the asymptotic behavior of precise large deviations of heavy-tailed random sums. The classical results in this context are due to Nagacv (1969a)(1969b). This paper includes three parts. First of all, we investigate the precise large deviations for a sum of independent but not identical distributed random variables.{Xn,n≥1} are independent non-negative random variables with distribution functions {Fn,n≥1}. We assume that the average of right tails of distribution functions Fn is equivalent to some distribution function F with consistently varying tails. Then we get the asymptotic behavior of the precise large deviations for a sum of independent but not identical distributed random variables, and we applied our results to a realistic example (Pareto-type distribution) and obtain a specific result. Secondly, we extend the precise large deviations in the compound renewal risk model. Compared with the result in Kon-stantinides and Loukissas (2010), we give the same result under weaker condition. In addition, we also give another result which extends Theorem2.4in Tang et al.(2001) from ERV{-α,-β) to C class. The method proofs are based on early works of Tang and Cline et al. Finally, according to the similarity in theory between the tail probability and the large deviation, we also get the asymptotic behavior of the tail probabilities in a new different condition.