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Weak convergence to the coalescent in neutral population models

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

M. Möhle
M. Möhle*
Affiliation:
University of Oxford and Johannes Gutenberg-Universität Mainz
*
∗Postal address: The University of Oxford, Department of Statistics, 1 South Parks Road, Oxford OX1 3TG, UK. and (2) Department of Mathematics, Johannes Gutenberg-Universität Mainz, Fachbereich Mathematik, Saarstraße 21, 55099 Mainz, Germany. Email address: (1) moehle@stats.ox.ac.uk, (2) moehle@mathematik.uni-mainz.de.
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Abstract

For a large class of neutral population models the asymptotics of the ancestral structure of a sample of n individuals (or genes) is studied, if the total population size becomes large. Under certain conditions and under a well-known time-scaling, which can be expressed in terms of the coalescence probabilities, weak convergence in DE([0,∞)) to the coalescent holds. Further the convergence behaviour of the jump chain of the ancestral process is studied. The results are used to approximate probabilities which are of certain interest in applications, for example hitting probabilities.

Keywords

Coalescence probabilityhitting probabilityjump chainneutralitygenealogical processpopulation geneticsrobustnessweak convergence

MSC classification

Primary: 60F05: Central limit and other weak theorems 92D10: Genetics
Secondary: 60G35: Signal detection and filtering 92D25: Population dynamics (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 446 - 460
Copyright
Copyright © Applied Probability Trust 1999 

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References

Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Cannings, C. (1974). The latent roots of certain Markov chains arising in genetics: a new approach, I. Haploid models. Adv. Appl. Prob. 6, 260–290.CrossRefGoogle Scholar
Donnelly, P. (1991). Weak convergence to a Markov chain with an entrance boundary: ancestral processes in population genetics. Ann. Prob. 19, 1102–1117.CrossRefGoogle Scholar
Donnelly, P. and Tavaré, S. (1995). Coalescents and genealogical structure under neutrality. Ann. Rev. Genet. 29, 401–421.CrossRefGoogle ScholarPubMed
Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes, Characterization and Convergence. Wiley, New York.CrossRefGoogle Scholar
Herbots, H. M. (1994) Stochastic models in population genetics: genealogy and genetic differentiation in structured populations. , Queen Mary and Westfield College, University of London.Google Scholar
Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science, ed. Gani, J. and Hannon, E. J. (J. Appl. Prob. 19A). Applied Probability Trust, Sheffield, UK, pp. 27–43.Google Scholar
Kingman, J. F. C. (1982). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, eds. Koch, G. and Spizzichino, F. North–Holland, New York, pp. 97–112.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Proc. Appl. 13, 235–248.CrossRefGoogle Scholar
Möhle, M. (1998) Robustness results for the coalescent. J. Appl. Prob 35, 438–447.CrossRefGoogle Scholar
Tavaré, S. (1984). Line-of-descent and genealogical processes, and their applications in population genetics models. Theor. Pop. Biol. 26, 119–164.CrossRefGoogle ScholarPubMed