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Bisexual Galton–Watson branching processes with superadditive mating functions

Published online by Cambridge University Press:  14 July 2016

D. J. Daley ,
David M. Hull  and
James M. Taylor
D. J. Daley*
Affiliation:
The Australian National University
David M. Hull*
Affiliation:
Valparaiso University
James M. Taylor
Affiliation:
The Australian National University
*
∗Postal address: Statistics Department, (IAS), Australian National University, GPO Box 4, Canberra, ACT 2601, Australia.
∗∗Postal address: Department of Mathematics and Computer Science, Valparaiso University, Valparaiso, IN 46383, USA.
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Abstract

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

Keywords

EXTINCTION PROBABILITYOFFSPRING DISTRIBUTION
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 3 , September 1986 , pp. 585 - 600
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

∗∗∗

Present address: 73 Torrington Road, Maroubra, NSW 2035, Australia.

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