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. Poincaré Probab. Statist. 51, 867-900 (2015)] and [Stoch. Process. 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 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.
Funding Statement
C. J. acknowledges support from National Natural Science Foundation of China with grant No. U2230402 and grant No. 12271020. D.-Q. J. acknowledges support from National Natural Science Foundation of China with grant No. 11871079 and grant No. 12090015.
Acknowledgments
We thank Prof. Xian Chen and Prof. Youming Li for stimulating discussions. We are also grateful to the anonymous referees for their helpful comments and suggestions. C. J. acknowledges support from National Natural Science Foundation of China with grant No. U2230402 and grant No. 12271020. D.-Q. J. acknowledges support from National Natural Science Foundation of China with grant No. 11871079 and grant No. 12090015.
Citation
Chen Jia. Da-Quan Jiang. Bingjie Wu. "Large deviations for the empirical measure and empirical flow of Markov renewal processes with a countable state space." Electron. J. Probab. 29 1 - 49, 2024. https://doi.org/10.1214/24-EJP1103
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