Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims

In this paper, we establish the second order asymptotics of ruin probabilities of a renewal risk model under the condition that the equilibrium distribution of claim sizes belongs to a rather general heavy-tailed distribution subclass—the class of second order subexponential distributions with finite mean. What is more, this requirement is proved to be necessary. Furthermore, a rather general sufficient condition on the claim size distribution itself is presented. Moreover, an extension to the case of random walk is also included.