Large deviations for renewal processes

TL;DR

This paper studies large deviations in renewal processes, revealing a unique rate functional that differs from classical theories, especially for arbitrary waiting-time distributions.

Contribution

It introduces a novel large deviations rate functional for renewal processes that diverges from Donsker-Varadhan theory, applicable to arbitrary waiting-time distributions.

Findings

The rate functional is non-strictly convex.

The rate functional is non-analytic.

Classical large deviations theory does not apply here.

Abstract

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.