A Note on the Regularity Property of Semi-Markov Processes with Borel State Space

This note shows that a semi-Markov process with Borel state space is under a fairly weak condition on the or and assuming that the embedded Markov chain satisfies either one of the following conditions: (a) it is ; (b) it is and the “recurrent part” of the state space is reached with probability one for every initial state; (c) it has a Under the latter condition, the regularity property is only ensured for almost all initial state with respect to the invariant probability measure