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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.

References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington.Google Scholar
Asmussen, S. and Hering, H. (1983) Branching Processes. Birkhäuser, Boston.Google Scholar
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Bingham, N. H. and Doney, R. A. (1974) Asymptotic properties of supercritical branching processes I: The Galton–Watson process. Adv. Appl. Prob. 6, 711–731.Google Scholar
Dubuc, S. (1970) La fonction de Green d'un processus de Galton–Watson. Studia Math. 34, 69–87.Google Scholar
Ewens, W. J. (1979) Mathematical Population Genetics. Springer-Verlag, Berlin.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. Wiley, New York.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Karlin, S. and Mcgregor, J. (1967) The number of mutant forms maintained in a population. Proc. Fifth Berk. Symp. Math. Statist. Prob. 4, 415–438.Google Scholar
Kingman, J. F. C. (1980) Mathematics of Genetic Diversity. SIAM, Philadelphia.CrossRefGoogle Scholar
Makarov, G. D. (1980) Large deviations for a critical Galton–Watson process. Theory Prob. Appl. 25, 481–492.CrossRefGoogle Scholar
Pakes, A. G. (1971) Some limit theorems for the total progeny of a branching process. Adv. Appl. Prob. 3, 176–192.Google Scholar
Pakes, A. G. (1973) Conditional limit theorems for a left-continuous random walk. J. Appl. Prob. 10, 39–53.Google Scholar
Pakes, A. G. (1978) On the age distribution of a Markov chain. J. Appl. Prob. 15, 65–77.Google Scholar
Pakes, A. G. (1988) An infinite alleles version of the Markov branching process. J. Austral. Math. Soc. To appear.Google Scholar
Seneta, E. (1967) The Galton–Watson process with mean one. J. Appl. Prob. 4, 489–495.CrossRefGoogle Scholar