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An infinite-alleles version of the simple branching process

Published online by Cambridge University Press:  01 July 2016

R. C. Griffiths  and
Anthony G. Pakes
R. C. Griffiths*
Affiliation:
Monash University
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
∗Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
∗∗Postal address: Department of Mathematics. The University of Western Australia, Nedlands, WA 6009, Australia.
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Abstract

Individuals in a population which grows according to the rules defining the simple branching process can mutate to novel allelic forms. We obtain limit theorems for the number of alleles present in any generation, the total number of alleles ever seen and the number of the generation containing the last mutation event.

In addition we define a notion of frequency spectrum for each generation as the expected number of alleles having a given number of representatives. As the generation number increases we prove the existence of a limiting notion of the frequency spectrum and discuss its upper tail behaviour. Our results here are incomplete and we make some conjectures which are supported by informal argument and specific examples.

Keywords

NEUTRAL ALLELESPOPULATION GENETICSFREQUENCY SPECTRUM
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 3 , September 1988 , pp. 489 - 524
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research carried out at Colorado State University.

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