Empirical estimator of stationary distribution for semi-Markov processes

The problem of statistical inference for semi-Markov processes is of increasing interest in recent literature. The aim of this article is to present an empirical estimator of the stationary distribution for semi-Markov processes. We use the empirical estimators for the stationary distribution of the embedded Markov chain and for the mean sojourn time. The main results given here are the asymptotic properties of these estimators, as the strong consistency and the asymptotic normality.