### Abstract

We introduce the concept of asymptotic period for an irreducible and aperiodic, discrete-time Markov chain X on a countable state space, and develop the theory leading to its formal definition. The asymptotic period of X equals one - its period - if X is recurrent, but may be larger than one if X is transient; X is asymptotically aperiodic if its asymptotic period equals one. Some sufficient conditions for asymptotic aperiodicity are presented. The asymptotic period of a birth-death process on the nonnegative integers is studied in detail and shown to be equal to 1, 2 or infinity. Criteria for the occurrence of each value in terms of the 1-step transition probabilities are established.

Original language | English |
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Pages (from-to) | 759-778 |

Number of pages | 20 |

Journal | Markov processes and related fields |

Volume | 24 |

Issue number | 5 |

Publication status | Published - 2018 |

### Keywords

- aperiodicity
- birth-death process
- harmonic function
- period
- transient Markov chain
- transition probability

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## Cite this

van Doorn, E. A. (2018). Asymptotic period of an aperiodic Markov chain.

*Markov processes and related fields*,*24*(5), 759-778.