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Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 84640, 15 pages
http://dx.doi.org/10.1155/JAMSA/2006/84640

Quasi-stationary distributions for birth-death processes with killing

1Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK
2Department of Applied Mathematics, University of Twente, P.O. Box 217, Enschede 7500, AE, The Netherlands

Received 9 January 2006; Revised 9 June 2006; Accepted 28 July 2006

Copyright © 2006 Pauline Coolen-Schrijner and Erik A. van Doorn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Karlin-McGregor representation for the transition probabilities of a birth-death process with an absorbing bottom state involves a sequence of orthogonal polynomials and the corresponding measure. This representation can be generalized to a setting in which a transition to the absorbing state (killing) is possible from any state rather than just one state. The purpose of this paper is to investigate to what extent properties of birth-death processes, in particular with regard to the existence of quasi-stationary distributions, remain valid in the generalized setting. It turns out that the elegant structure of the theory of quasi-stationarity for birth-death processes remains largely intact as long as killing is possible from only finitely many states. In particular, the existence of a quasi-stationary distribution is ensured in this case if absorption is certain and the state probabilities tend to zero exponentially fast.