Large deviations for the empirical measure and empirical flow of Markov renewal processes with a countable state space

TL;DR

This paper establishes large deviation principles for the empirical measure and flow of Markov renewal processes with countable states, extending previous results and providing explicit rate functions under certain conditions.

Contribution

It generalizes large deviation results to Markov renewal processes with countable state spaces, weakening compactness conditions and deriving explicit rate functions.

Findings

Proves joint large deviation principle for empirical measure and flow.

Extends results to countable state space Markov renewal processes.

Provides explicit marginal rate functions under stronger conditions.

Abstract

Here we propose the Donsker-Varadhan-type compactness conditions and prove the joint large deviation principle for the empirical measure and empirical flow of Markov renewal processes (semi-Markov processes) with a countable state space, generalizing the relevant results for continuous-time Markov chains with a countable state space obtained in [Ann. Inst. H. Poincar\'{e} Probab. Statist. 51, 867-900 (2015)] and [Stoch. Proc. Appl. 125, 2786-2819 (2015)], as well as the relevant results for Markov renewal processes with a finite state space obtained in [Adv. Appl. Probab. 48, 648-671 (2016)]. In particular, our results hold when the flow space is endowed with either the bounded weak* topology or the strong $L^1$ topology. Even for continuous-time Markov chains, our compactness conditions are weaker than the ones proposed in previous papers. Furthermore, under some stronger conditions, we obtain the explicit expression of the marginal rate function of the empirical flow.