Extinction probability in a birth-death process with killing. (English) Zbl 1083.60071
Authors’ abstract: We study birth-death processes on the nonnegative integers, where \(\{1,2,\dots\}\) 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 \(t\to\infty\), 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.
Reviewer: Valentin Topchii (Omsk)
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 60J27 | Continuous-time Markov processes on discrete state spaces |
Keywords:
absorption; decay parameter; extinction time; persistence time; rate of convergence; logarithmic normReferences:
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