### Abstract

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.

Original language | English |
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Title of host publication | Systems and Control |

Subtitle of host publication | Foundations and Applications |

Publisher | Birkhauser Verlag Basel |

Pages | 301-309 |

Number of pages | 9 |

Edition | 9780817683368 |

DOIs | |

State | Published - 1 Jan 2012 |

### Publication series

Name | Systems and Control: Foundations and Applications |
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Number | 9780817683368 |

ISSN (Print) | 2324-9749 |

ISSN (Electronic) | 2324-9757 |

### Keywords

- Borel Space
- Embed Markov Chain
- Invariant Probability Measure
- Markov Chain
- Regularity Property

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## Cite this

Vega-Amaya, Ó. (2012). On the regularity property of Semi-Markov processes with Borel state spaces. In

*Systems and Control: Foundations and Applications*(9780817683368 ed., pp. 301-309). (Systems and Control: Foundations and Applications; No. 9780817683368). Birkhauser Verlag Basel. https://doi.org/10.1007/978-0-8176-8337-5_18