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The age of a Galton-Watson population with a geometric offspring distribution

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

F. C. Klebaner  and
S. Sagitov
F. C. Klebaner*
Affiliation:
Monash University
S. Sagitov*
Affiliation:
Chalmers University of Technology
*
∗ Postal address: School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia.
∗∗ Postal address: School of Mathematical Sciences, Chalmers University of Technology, S-412 96 Gothenburg, Sweden. Email address: serik@math.chalmers.se
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Abstract

Motivated by the question of the age in a branching population we try to recreate the past by looking back from the currently observed population size. We define a new backward Galton-Watson process and study the case of the geometric offspring distribution with parameter p in detail. The backward process is then the Galton-Watson process with immigration, again with a geometric offspring distribution but with parameter 1-p, and it is also the dual to the original Galton-Watson process. We give the asymptotic distribution of the age when the initial population size is large in supercritical and critical cases. To this end, we give new asymptotic results on the Galton-Watson immigration processes stopped at zero.

Keywords

Backward Galton-Watson processpopulation agegeometric offspring distributionrandom pastimmigration stopped at zerocritical

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 816 - 828
Copyright
Copyright © Applied Probability Trust 2002 

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