Quasi-stationary Distributions for a Class of Discrete-time Markov Chains

Pauline Coolen-Schrijner () and Erik A. Doorn ()
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Pauline Coolen-Schrijner: Durham University
Erik A. Doorn: University of Twente

Methodology and Computing in Applied Probability, 2006, vol. 8, issue 4, 449-465

Abstract: 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 (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.

Keywords: Absorption probability; Birth–death process with killing; Decay parameter; Quasi-stationarity; Rate of convergence; Primary 60J80; Secondary 60J10 (search for similar items in EconPapers)
Date: 2006
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11009-006-0424-y

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