Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-11T15:01:19.191Z Has data issue: false hasContentIssue false
  • English
  • Français

A convergence theorem for markov chains arising in population genetics and the coalescent with selfing

Part of: Limit theorems

Published online by Cambridge University Press:  01 July 2016

M. Möhle
M. Möhle*
Affiliation:
University of Chicago and Johannes Gutenberg-Universität Mainz
*
∗ Postal address: (1) The University of Chicago, Department of Statistics, 5734 University Avenue, Chicago, IL 60637, USA, (2) Johannes Gutenberg-Universität Mainz, Fachbereich Mathematik, Saarstraße 21, 55099 Mainz, Germany. Email address: (1)[email protected], (2) [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A simple convergence theorem for sequences of Markov chains is presented in order to derive new ‘convergence-to-the-coalescent’ results for diploid neutral population models.

For the so-called diploid Wright-Fisher model with selfing probability s and mutation rate θ, it is shown that the ancestral structure of n sampled genes can be treated in the framework of an n-coalescent with mutation rate ̃θ := θ(1-s/2), if the population size N is large and if the time is measured in units of (2-s)N generations.

Keywords

Coalescentdiploid population modelsgenealogical processpopulation geneticsrobustnessselfing

MSC classification

Primary: 60F05: Central limit and other weak theorems 92D10: Genetics
Secondary: 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 493 - 512
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Brown, A. H. D. (1990). Genetic characterization of plant mating systems. In Plant Population Genetics, Breeding, and Genetic Resources, eds. Brown, A. H. D., Clegg, M. T., Kahler, A. L., Weir, B. S.. Sinauer Associates Inc., Sunderland, Massachusetts, pp. 145–162.Google Scholar
[2] Clegg, M. T. (1980). Measuring plant mating systems. BioScience 30, 814–818.CrossRefGoogle Scholar
[3] Donnelly, P. and Tavaré, S. (1995). Coalescents and genealogical structure under neutrality. A. Rev. Genet. 29, 401–421.CrossRefGoogle ScholarPubMed
[4] Griffiths, R. C. and Majoram, P. (1997). An ancestral recombination graph. In Progress in Population Genetics and Human Evolution, ed. Donnelly, P.. Springer, pp. 257–270.CrossRefGoogle Scholar
[5] Hudson, R. R. and Kaplan, N. L. (1988). The coalescent process in models with selection and recombination. Genetics 120, 831–840.CrossRefGoogle ScholarPubMed
[6] Kingman, J. F. C. (1982). On the genealogy of large populations. J. Appl. Prob. 19A, 27–43.CrossRefGoogle Scholar
[7] Kingman, J. F. C. (1982). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, eds. Koch, G., F. Spizzichino. North-Holland Publishing Company, pp. 97–112.Google Scholar
[8] Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
[9] Milligan, B. G. (1996). Estimating long-term mating systems using DNA sequences. Genetics 142, 619–627.Google Scholar
[10] Möhle, M., Robustness results for the coalescent. J. Appl. Prob. 35, 437–446.Google Scholar
[11] Möhle, M., Coalescent results for two-sex population models. Adv. Appl. Prob. 30, 513–520.Google Scholar
[12] Nordborg, M. and Donnelly, P. The coalescent process with selfing. Genetics 146, 1185–1195.CrossRefGoogle Scholar
[13] Pollak, E. (1987). On the theory of partially inbreeding finite populations. I. Partial selfing. 117, 353–360.Google ScholarPubMed
[14] Tavaré, S., (1984). Line-of-descent and genealogical processes, and their applications in population genetics models. Theor. Pop. Biol. 26, 119–164.CrossRefGoogle ScholarPubMed
[15] Wright, S. (1969). Evolution and the Genetics of Populations. Volume 2. University of Chicago Press, Chicago.Google Scholar