Mikael Petersson: Asymptotic expansions for perturbed discrete time renewal equations

We consider a discrete time renewal equation depending on a small perturbation parameter and study the asymptotic behaviour of the solution. Under Cramer type conditions, an exponential asymptotic expansion of the solution is obtained. The results are applied to the study of quasi-stationary phenomena for perturbed regenerative processes. We define quasi-stationary distributions for such processes and study their asymptotic properties. In particular, we build an asymptotic power series expansion of the quasi-stationary distribution with respect to the perturbation parameter. Finally, it is described how the results can be used to obtain approximations of ruin probabilities for perturbed discrete time risk processes.