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Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

Published online by Cambridge University Press:  01 July 2016

Erik A. Van Doorn
Erik A. Van Doorn*
Affiliation:
University of Twente
*
∗Postal address: Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
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Abstract

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

Keywords

DECAY PARAMETERDUALITYORTHOGONAL POLYNOMIALSQUASI-LIMITINGDISTRIBUTIONQUASI-STATIONARY DISTRIBUTIONRATE OF CONVERGENCESPECTRAL REPRESENTATION
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 4 , December 1991 , pp. 683 - 700
Copyright
Copyright © Applied Probability Trust 1991 

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