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A Galton–Watson process with a threshold

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  21 June 2016

K. B. Athreya  and
H.-J. Schuh
K. B. Athreya*
Affiliation:
Iowa State University
H.-J. Schuh*
Affiliation:
Johannes Gutenberg-Universität
*
* Postal address: Departments of Mathematics and Statistics, Iowa State University, Ames, IA 50010, USA. Email address: kba@iastate.edu
** Postal address: Fachbereich 8, Physik, Mathematik und Informatik, Johannes Gutenberg-Universität, D-55099 Mainz, Germany. Email address: schuh@mathematik.uni-mainz.de
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Abstract

In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

Keywords

Branching processsize dependencethresholdextinction time

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations
Type
Short Communications
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 614 - 621
Copyright
Copyright © Applied Probability Trust 2016 

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References

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