Large deviations for the empirical measure of heavy tailed Markov renewal processes

TL;DR

This paper establishes a large deviations principle for renewal Markov processes with heavy-tailed distributions, revealing unique degeneracies and non-convexities in the rate functional that differ from classical expectations.

Contribution

It introduces a large deviations framework for heavy-tailed renewal Markov processes without bounded waiting times, highlighting novel degeneracies in the rate functional.

Findings

Rate functional is degenerate with a nontrivial set of zeros

The rate functional is not strictly convex

Behavior differs from heuristic Donsker-Varadhan analysis

Abstract

A large deviations principle is established for the joint law of the empirical measure and the flow measure of a renewal Markov process on a finite graph. We do not assume any bound on the arrival times, allowing heavy tailed distributions. In particular, the rate functional is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behavior highly different from what one may guess with a heuristic Donsker-Varadhan analysis of the problem.