Hostname: page-component-cb9f654ff-qc88w Total loading time: 0 Render date: 2025-08-22T20:21:30.736Z Has data issue: false hasContentIssue false
  • English
  • Français

A Galton–Watson process with a threshold

Part of: Limit theorems Markov processes

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
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 614 - 621
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Athreya, K. B. and Ney, P. E. (1972).Branching Processes.Springer, New York.Google Scholar
[2]Eisenberg, B. (2008).On the expectation of the maximum of IID geometric random variables.Statist. Prob. Lett. 78, 135–143.Google Scholar
[3]Jagers, P. (1996).Population-size-dependent branching processes.J. Appl. Math. Stoch. Anal. 9, 449–457.Google Scholar
[4]Klebaner, F. C. (1984).Geometric rate of growth in population size dependent branching processes.J. Appl. Prob. 21, 40–49.CrossRefGoogle Scholar
[5]Klebaner, F. C. (2010).Approximation in population-size-dependent branching processes. In Workshop on branching processes and their applications (Lecture Notes Statist. Proc.197), pp.71–78.Google Scholar
[6]Seneta, E. (1967).The Galton–Watson process with mean one.J. Appl. Prob. 4, 489–495.Google Scholar
[7]Slack, E. (1968).A branching process with mean one and possibly infinite variance.Z. Wahrscheinlichkeitsth. 9, 139–145.CrossRefGoogle Scholar