Some Algebraic Methods in Semi-Markov Chains

This article is concerned with algebraic methods developed for discrete-time semi-Markov processes with countable state space. After a short presentation of infinite matrices and associated operations, we describe the discrete-time semi-Markov setting. Then, we present some elements of Markov renewal theory and we obtain the mean hitting time of a subset of the state space for this type of process. A specialization of these results to the finite state space case is also presented. We end with several concluding remarks.