Quasi-stationary distributions for a class of discrete-time Markov chains

P. Coolen-Schrijner, E.A. van Doorn

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Abstract

This paper is concerned with the circumstances under which a discrete-time absorbing Markov chain has a quasi-stationary distribution. We showed in a previous paper that a pure birth-death process with an absorbing bottom state has a quasi-stationary distribution -- actually an infinite family of quasi-stationary distributions -- if and only if absorption is certain and the chain is geometrically transient. If we widen the setting by allowing absorption in one step ({\it killing}) from any state, the two conditions are still necessary, but no longer sufficient. We show that the birth-death-type of behaviour prevails as long as the number of states in which killing can occur is finite. But if there are infinitely many such states, and if the chain is geometrically transient and absorption certain, then there may be 0, 1, or infinitely many quasi-stationary distributions. Examples of each type of behaviour are presented. We also survey and supplement the theory of quasi-stationary distributions for discrete-time Markov chains in general.
Original languageEnglish
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages22
Publication statusPublished - 2004

Publication series

NameMemorandum Faculty of Mathematical Sciences
PublisherUniversity of Twente, Department of Applied Mathematics
No.1737
ISSN (Print)0169-2690

Fingerprint

Quasi-stationary Distribution
Markov chain
Discrete-time
Absorption
Absorbing
Birth-death Process
Class
Sufficient
If and only if
Necessary

Keywords

  • MSC-60J10
  • EWI-3557
  • METIS-219033
  • IR-65921
  • MSC-60J80

Cite this

Coolen-Schrijner, P., & van Doorn, E. A. (2004). Quasi-stationary distributions for a class of discrete-time Markov chains. (Memorandum Faculty of Mathematical Sciences; No. 1737). Enschede: University of Twente, Department of Applied Mathematics.
Coolen-Schrijner, P. ; van Doorn, E.A. / Quasi-stationary distributions for a class of discrete-time Markov chains. Enschede : University of Twente, Department of Applied Mathematics, 2004. 22 p. (Memorandum Faculty of Mathematical Sciences; 1737).
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Coolen-Schrijner, P & van Doorn, EA 2004, Quasi-stationary distributions for a class of discrete-time Markov chains. Memorandum Faculty of Mathematical Sciences, no. 1737, University of Twente, Department of Applied Mathematics, Enschede.

Quasi-stationary distributions for a class of discrete-time Markov chains. / Coolen-Schrijner, P.; van Doorn, E.A.

Enschede : University of Twente, Department of Applied Mathematics, 2004. 22 p. (Memorandum Faculty of Mathematical Sciences; No. 1737).

Research output: Book/ReportReportProfessional

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N2 - This paper is concerned with the circumstances under which a discrete-time absorbing Markov chain has a quasi-stationary distribution. We showed in a previous paper that a pure birth-death process with an absorbing bottom state has a quasi-stationary distribution -- actually an infinite family of quasi-stationary distributions -- if and only if absorption is certain and the chain is geometrically transient. If we widen the setting by allowing absorption in one step ({\it killing}) from any state, the two conditions are still necessary, but no longer sufficient. We show that the birth-death-type of behaviour prevails as long as the number of states in which killing can occur is finite. But if there are infinitely many such states, and if the chain is geometrically transient and absorption certain, then there may be 0, 1, or infinitely many quasi-stationary distributions. Examples of each type of behaviour are presented. We also survey and supplement the theory of quasi-stationary distributions for discrete-time Markov chains in general.

AB - This paper is concerned with the circumstances under which a discrete-time absorbing Markov chain has a quasi-stationary distribution. We showed in a previous paper that a pure birth-death process with an absorbing bottom state has a quasi-stationary distribution -- actually an infinite family of quasi-stationary distributions -- if and only if absorption is certain and the chain is geometrically transient. If we widen the setting by allowing absorption in one step ({\it killing}) from any state, the two conditions are still necessary, but no longer sufficient. We show that the birth-death-type of behaviour prevails as long as the number of states in which killing can occur is finite. But if there are infinitely many such states, and if the chain is geometrically transient and absorption certain, then there may be 0, 1, or infinitely many quasi-stationary distributions. Examples of each type of behaviour are presented. We also survey and supplement the theory of quasi-stationary distributions for discrete-time Markov chains in general.

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Coolen-Schrijner P, van Doorn EA. Quasi-stationary distributions for a class of discrete-time Markov chains. Enschede: University of Twente, Department of Applied Mathematics, 2004. 22 p. (Memorandum Faculty of Mathematical Sciences; 1737).