Higher-order Expansions for Compound Distributions and Ruin Probabilities with Subexponential Claims

Let Xi (i = 1, 2, . . .) be a sequence of subexponential positive independent and identically distributed random variables. In this paper we offer two alternative approaches to obtain higher-order expansions of the tail of ∑n i=1 Xi and subsequently for ruin probabilities in renewal risk models with claim sizes Xi. In particular, these emphasize the importance of the term P( ∑n i=1 Xi > s,max(X1, . . . , Xn) ≤ s/2) for the accuracy of the resulting asymptotic expansion of P( ∑n i=1 Xi > s). Furthermore, we present a more rigorous approach to the often suggested technique of using approximations with shifted arguments. The cases of a Pareto-type, Weibull and Lognormal distribution for Xi are discussed in more detail and numerical investigations of the increase in accuracy by including higher-order terms in the approximation of ruin probabilities for finite realistic ranges of s are given.