Nonparametric Estimation of the Stationary Distribution of a Discrete-Time Semi-Markov Process

In this article, we consider a discrete-time semi-Markov process with finite state space and an observation censored at an arbitrary fixed time. Some intermediate results concerning the empirical estimation of the mean recurrence times of the embedded Markov process and the mean sojourn times of the semi-Markov process are given. We study two nonparametric estimators for the stationary distribution of the semi-Markov process and examine their asymptotic properties, such as strong consistency and asymptotic normality, as the length of the observation tends to infinity. Finally, a numerical application is presented to illustrate the comparison of the two estimators.