On the Regularity Property of Semi-Markov Processes with Borel State Spaces
This note shows that a semi-Markov process with Borel state space is regular under a fairly weak condition on the mean sojourn or holding times and assuming that the embedded Markov chain satisfies one of the following conditions: (a) it is Harris recurrent; (b) it is recurrent and the “recurrent part” of the state space is reached with probability one for every initial state; (c) it has a unique invariant probability measure. Under the latter condition, the regularity property is only ensured for almost all initial states with respect to the invariant probability measure.
KeywordsMarkov Chain Regularity Property Invariant Probability Measure Borel Space Embed Markov Chain
The author takes this opportunity to thank to Professor Onésimo Hernández-Lerma for his constant encouragement and support during the author’s academic career and, in particular, for his valuable comments on an early version of present work. Thanks are also due to the referees for their useful observations which improve the writing and organization of this note.
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