A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker‒Varadhan analysis of the problem.
"Large deviations for the empirical measure of heavy-tailed Markov renewal processes." Adv. in Appl. Probab. 48 (3) 648 - 671, September 2016.