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A Pólya urn model and the coalescent

Part of: Combinatorial probability Markov processes

Published online by Cambridge University Press:  14 July 2016

Gudrun Trieb
Gudrun Trieb*
Affiliation:
Johannes-Gutenberg-Universität, Mainz
*
∗ Postal address: Fachbereich Mathematik, Johannes-Gutenberg-Universität, DW-6500 Mainz, Germany.
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Abstract

In recent papers by Hoppe and Donnelly it has been shown that a Pólya urn model generating the Ewens sampling formula (population genetics) parallels a construction of Kingman using a Poisson–Dirichlet ‘paintbox'. Even the jump chain of Kingman's n-coalescent can be constructed using the urn. The properties of a certain process based on the coalescent also are derived. This process was introduced by Hoppe.

Keywords

EWENS SAMPLING FORMULAGENEALOGYMUTATIONPOPULATION GENETICS

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60C05: Combinatorial probability 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 92D10: Genetics
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 1 - 10
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1]Donnelly, P. J. (1986) Partition structures, Pólya urns, the Ewens sampling formula and the ages of alleles. Theoret. Popn. Biol. 30, 271–288.Google Scholar
[2]Donnelly, P. J. and Tavare, S. (1986) The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19.Google Scholar
[3]Donnelly, P. J. and Tavaré, S. (1987) The population genealogy of the infinitely-many neutral alleles model. J. Math. Biol. 25, 381–391.Google Scholar
[4]Donnelly, P. J. and Tavaré, S. (1987) A genealogical description of the infinitely-many neutral alleles model. In Stochastic Methods in Biology, ed. Kimura, M., Kallianpur, C. and Hida, T., Lecture Notes in Biomathematics 70, pp. 27–35, Springer-Verlag, Berlin.Google Scholar
[5]Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn. Biol. 3, 87–112.Google Scholar
[6]Ewens, W. J. (1979) Mathematical Population Genetics. Springer-Verlag, Berlin.Google Scholar
[7]Hoppe, F. M. (1984) Pólya-like urns and the Ewens sampling formula. J. Math. Biol. 20, 91–94.Google Scholar
[8]Hoppe, F. M. (1987) The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25, 123–159.Google Scholar
[9]Karlin, S. and Mcgregor, J. (1972) Addendum to a paper of W. Ewens. Theoret. Popn. Biol. 3, 113–116.Google Scholar
[10]Kingman, J. F. C. (1977) The population structure associated with the Ewens sampling formula. Theoret. Popn. Biol. 11, 274–283.Google Scholar
[11]Kingman, J. F. C. (1978) Random partitions in population genetics. Proc. R. Soc. London A361, 1–20.Google Scholar
[12]Kingman, J. F. C. (1978) The representation of partition structures. J. Lond. Math. Soc. (2) 18, 374–380.Google Scholar
[13]Kingman, J. F. C. (1982) On the genealogy of large populations. J. Appl. Prob. 19A, 27–43.Google Scholar
[14]Kingman, J. F. C. (1982) The coalescent. Stoch. Proc. Appl. 13, 235–248.Google Scholar
[15]Kingman, J. F. C. (1982) Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, pp. 97–112. North-Holland, Amsterdam.Google Scholar
[16]Watterson, G. A. (1974) The sampling theory of selectively neutral alleles. Adv. Appl. Prob. 6, 463–488.Google Scholar
[17]Watterson, G. A. (1974) Models for the logarithmic series abundance distributions. Theoret. Popn. Biol. 6, 217–250.Google Scholar
[18]Watterson, G. A. (1976) The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.Google Scholar
[19]Watterson, G. A. (1984) Lines of descent and the coalescent. Theoret. Popn. Biol. 26, 77–92.Google Scholar