Precise large deviation for sums of sub-exponential claims with the [formula omitted]-dependent semi-Markov type structure

Abstract

Consider a renewal risk model in which the current inter-arrival time depends on a fixed number of previous claims but is independent of all other claims. We are interested in the precise large deviations of the aggregate amount of claims for the case of sub-exponential claims. In addition, an asymptotic estimate for the expectation of the aggregate amount of claims is also obtained. The asymptotic formulas are fully consistent with existing works in this study.