Metastable distributions of semi-Markov processes

In this paper, we consider semi-Markov processes whose transition times and transition probabilities depend on a small parameter [Formula: see text]. We are interested in the behavior of a process [Formula: see text] at times [Formula: see text] that depend on the value of the parameter. This behavior depends on how the point [Formula: see text] approaches infinity. We introduce the notion of complete asymptotic regularity (a certain asymptotic condition on transition probabilities and transition times), originally developed for parameter-dependent Markov chains, which ensures the existence of the metastable distribution for each initial point and a given time scale [Formula: see text]. The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes.