Asymptotic period of an aperiodic Markov chain

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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 languageEnglish
Pages (from-to)759-778
Number of pages20
JournalMarkov processes and related fields
Volume24
Issue number5
Publication statusPublished - 2018

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Markov processes
Markov chain
Birth-death Process
Transition Probability
Countable
State Space
Discrete-time
Non-negative
Infinity
Integer
Sufficient Conditions

Keywords

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

Cite this

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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.",
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Asymptotic period of an aperiodic Markov chain. / van Doorn, Erik Alexander.

In: Markov processes and related fields, Vol. 24, No. 5, 2018, p. 759-778.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Asymptotic period of an aperiodic Markov chain

AU - van Doorn, Erik Alexander

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - aperiodicity

KW - birth-death process

KW - harmonic function

KW - period

KW - transient Markov chain

KW - transition probability

M3 - Article

VL - 24

SP - 759

EP - 778

JO - Markov processes and related fields

JF - Markov processes and related fields

SN - 1024-2953

IS - 5

ER -