Precise large deviations of aggregate claims in a risk model with regression-type size-dependence

In this note, we further extend the renewal risk model and introduce a more practical regression-type dependence structure between inter-arrival times and claim sizes, which is described by the framework of a web Markov skeleton process. We investigate large deviations of the aggregate claims, and, for a case of heavy-tailed claims, we obtain a precise large-deviation formula of the aggregate claims under some assumption about regression-type size-dependence, which is consistent with existing ones in the literature.