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

Research output: Book/ReportReportOther research output

Abstract

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.
LanguageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages25
Publication statusPublished - Mar 2017

Publication series

NameMemorandum / Department of Applied Mathematics
PublisherUniversity 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

van Doorn, E. A. (2017). 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.
van Doorn, Erik A. / Asymptotic period of an aperiodic Markov chain and the strong ratio limit property. Enschede : University of Twente, Department of Applied Mathematics, 2017. 25 p. (Memorandum / Department of Applied Mathematics; 2059).
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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",
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van Doorn, EA 2017, 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.

Enschede : University of Twente, Department of Applied Mathematics, 2017. 25 p. (Memorandum / Department of Applied Mathematics; No. 2059).

Research output: Book/ReportReportOther 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

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van Doorn EA. Asymptotic period of an aperiodic Markov chain and the strong ratio limit property. Enschede: University of Twente, Department of Applied Mathematics, 2017. 25 p. (Memorandum / Department of Applied Mathematics; 2059).