Kernel smoothing of the stationary distribution for continuous time semi-Markov process

The purpose of this paper is to investigate nonparametric estimation of one of the most important characteristics of homogeneous continuous-time semi-Markov processes, namely, the stationary distribution of the process. The construction of our estimator is based on the empirical estimator of the stationary distribution of the embedded Markov chain (EMC) and the kernel estimators for the mean sojourn time of the semi-Markov process (SMP) as well as the mean recurrence times in the state i of the Markov renewal process (MRP). When the time of observations goes to infinity, we derive the asymptotic properties of the proposed estimators as uniform strong consistency and asymptotic normality. Finally, a numerical example illustrates the asymptotic properties of the proposed estimators.