Large deviations for renewal processes

We investigate large deviations for the empirical measure of the forward and backward recurrence time processes associated with a classical renewal process with arbitrary waiting-time distribution. The Donsker–Varadhan theory cannot be applied in this case, and indeed it turns out that the large deviations rate functional differs from the one suggested by such a theory. In particular, a non-strictly convex and non-analytic rate functional is obtained.

► We study large deviations for the empirical measure of renewal processes.
► The rate functional is not the one suggested by the Donsker–Varadhan theory.
► A non-strictly convex and non-analytic rate functional is obtained.