### Abstract

Original language | Undefined |
---|---|

Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 25 |

Publication status | Published - Mar 2017 |

### Publication series

Name | Memorandum / Department of Applied Mathematics |
---|---|

Publisher | University of Twente, Department of Applied Mathematics |

No. | 2059 |

ISSN (Print) | 1874-4850 |

### Keywords

- ratio limit
- invariant vector
- Invariant Measure
- transition probability
- transient Markov chain
- period
- IR-104041
- aperiodicity
- Birth-death process
- MSC-60J10
- EWI-27823
- MSC-60J80
- harmonic function

### Cite this

*Asymptotic period of an aperiodic Markov chain and the strong ratio limit property*. (Memorandum / Department of Applied Mathematics; No. 2059). Enschede: University of Twente, Department of Applied Mathematics.

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*Asymptotic period of an aperiodic Markov chain and the strong ratio limit property*. Memorandum / Department of Applied Mathematics, no. 2059, University of Twente, Department of Applied Mathematics, Enschede.

**Asymptotic period of an aperiodic Markov chain and the strong ratio limit property.** / van Doorn, Erik A.

Research output: Book/Report › Report › Other research output

TY - BOOK

T1 - Asymptotic period of an aperiodic Markov chain and the strong ratio limit property

AU - van Doorn, Erik A.

PY - 2017/3

Y1 - 2017/3

N2 - We introduce the concept of asymptotic period for an irreducible and aperiodic discrete-time Markov chain on a countable state space. If the chain is transient its asymptotic period may be larger than one. We present some sufficient conditions and, in the more restricted setting of birth-death processes, a necessary and sufficient condition for asymptotic aperiodicity. It is subsequently shown that a birth-death process has the strong ratio limit property if a related birth-death process is asymptotically aperiodic. In the general setting a similar statement is not true, but validity of the converse implication is posed as a conjecture.

AB - We introduce the concept of asymptotic period for an irreducible and aperiodic discrete-time Markov chain on a countable state space. If the chain is transient its asymptotic period may be larger than one. We present some sufficient conditions and, in the more restricted setting of birth-death processes, a necessary and sufficient condition for asymptotic aperiodicity. It is subsequently shown that a birth-death process has the strong ratio limit property if a related birth-death process is asymptotically aperiodic. In the general setting a similar statement is not true, but validity of the converse implication is posed as a conjecture.

KW - ratio limit

KW - invariant vector

KW - Invariant Measure

KW - transition probability

KW - transient Markov chain

KW - period

KW - IR-104041

KW - aperiodicity

KW - Birth-death process

KW - MSC-60J10

KW - EWI-27823

KW - MSC-60J80

KW - harmonic function

M3 - Report

T3 - Memorandum / Department of Applied Mathematics

BT - Asymptotic period of an aperiodic Markov chain and the strong ratio limit property

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -