Entropy rate and maximum entropy methods for countable semi-Markov chains

We are concerned with introducing entropy in the field of countable discrete-time semi-Markov process theory. We define the entropy of the finite distributions of the semi-Markov chain and obtain explicitly its entropy rate by extending the Shannon-McMillan-Breiman theorem to this class of non-stationary discrete-time processes. We also define the relative entropy rate between two semi-Markov chains. We then develop some maximum entropy methods for these processes.