Extinction probability in a birth-death process with killing
Erik A. Van Doorn and Alexander I. Zeifman
Source: J. Appl. Probab. Volume 42, Number 1 (2005), 185-198.
Abstract
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.
Primary Subjects: 60J80
Secondary Subjects: 60J27
Keywords: Absorption; decay parameter; extinction time; persistence time; rate of convergence; logarithmic norm
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Permanent link to this document: http://projecteuclid.org/euclid.jap/1110381380
Digital Object Identifier: doi:10.1239/jap/1110381380
Mathematical Reviews number (MathSciNet):
MR2144903
Zentralblatt MATH identifier:
1083.60071
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