This paper is a study of Doeblin Ratio limit laws, the weak and strong laws of large numbers, and the Central Limit theorem for Markov Renewal processes. A general definition of these processes is given in Section 2. The means and variances of random variables associated with recurrence times are computed in Section 4. When restricted to the special case of a Markov chain, certain of the results of Sections 5 and 6 strengthen known results.
"Limit Theorems for Markov Renewal Processes." Ann. Math. Statist. 35 (4) 1746 - 1764, December, 1964. https://doi.org/10.1214/aoms/1177700397