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The time back to the most recent common ancestor in exchangeable population models

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

M. Möhle
M. Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
*
∗ Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: martin.moehle@uni-tuebingen.de
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Abstract

A class of haploid population models with population size N, nonoverlapping generations and exchangeable offspring distribution is considered. Based on an analysis of the discrete ancestral process, we present solutions, algorithms and strong upper bounds for the expected time back to the most recent common ancestor which hold for arbitrary sample size n ∈ {1,…,N}. New insights into the asymptotic behaviour of the expected time back to the most recent common ancestor for large population size are presented relating the results to coalescent theory.

Keywords

Absorption timeancestral processcoalescentexchangeabilityMoran modelmost recent common ancestorWright-Fisher model

MSC classification

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 78 - 97
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Cannings, C. (1974). The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. 6, 260–290.Google Scholar
[2] Cannings, C. (1975). The latent roots of certain Markov chains arising in genetics: a new approach. II. Further haploid models. Adv. Appl. Prob. 7, 264–282.CrossRefGoogle Scholar
[3] Chang, J. T. (1999). Recent common ancestors of all present-day individuals. Adv. Appl. Prob. 31, 1002–1026.CrossRefGoogle Scholar
[4] Crow, J. F. and Kimura, M. (1970). An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
[5] Ewens, W. J. (1979). Mathematical Population Genetics. Springer, Berlin.Google Scholar
[6] Gladstien, K. (1976). Loss of alleles in a haploid population with varying environment. Theoret. Pop. Biol. 10, 383–394.CrossRefGoogle Scholar
[7] Gladstien, K. (1977). Haploid populations subject to varying environment: the characteristic values and the rate of loss of alleles. SIAM J. Appl. Math. 32, 778–783.CrossRefGoogle Scholar
[8] Gladstien, K. (1978). The characteristic values and vectors for a class of stochastic matrices arising in genetics. SIAM J. Appl. Math. 34, 630–642.CrossRefGoogle Scholar
[9] Johnson, N. L. and Kotz, S. (1969). Distributions in Statistics: Discrete Distributions. Houghton Mifflin, Boston, MA.Google Scholar
[10] 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, Amsterdam, pp. 97–112.Google Scholar
[11] Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
[12] Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
[13] Möhle, M., (1998). Robustness results for the coalescent. J. Appl. Prob. 35, 438–447.CrossRefGoogle Scholar
[14] Möhle, M., (1999). The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 5, 761–777.CrossRefGoogle Scholar
[15] Möhle, M., (2000). Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. Appl. Prob. 32, 983–993.CrossRefGoogle Scholar
[16] Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29, 1547–1562.CrossRefGoogle Scholar
[17] Ross, S. M. (2002). Probability Models for Computer Science. Academic Press, San Diego, CA.Google Scholar
[18] Sagitov, S. (2003). Convergence to the coalescent with simultaneous multiple mergers. J. Appl. Prob. 40, 839–854.CrossRefGoogle Scholar
[19] Walsh, B. (2001). Estimating the time to the most recent common ancestor for the Y chromosome or mito-chon-drial DNA for a pair of individuals. Genetics 158, 897–912.CrossRefGoogle ScholarPubMed