Asymptotic period of an aperiodic Markov chain

Erik Alexander van Doorn

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2 Citations (Scopus)
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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
Issue number5
Publication statusPublished - 2018


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

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