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Extinction Probability in A Birth-Death Process with Killing

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Erik A. Van Doorn  and
Alexander I. Zeifman
Erik A. Van Doorn*
Affiliation:
University of Twente
Alexander I. Zeifman*
Affiliation:
Vologda State Pedagogical University and Vologda Scientific Coordinate Centre of CEMI RAS
*
∗Postal address: Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. Email address: [email protected]
∗∗Postal address: Vologda State Pedagogical University, S. Orlova 6, Vologda, Russia. Email address: [email protected]
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Abstract

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We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

Keywords

Absorptiondecay parameterextinction timepersistence timerate of convergencelogarithmic norm

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 185 - 198
Copyright
© Applied Probability Trust 2005 

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