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Convergence to the coalescent in populations of substantially varying size

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Peter Jagers  and
Serik Sagitov
Peter Jagers*
Affiliation:
Chalmers University of Technology and Göteborg University
Serik Sagitov*
Affiliation:
Chalmers University of Technology and Göteborg University
*
∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden.
∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden.
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Abstract

Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical trees of random samples of individuals in large but time-constant populations. Time is viewed as being discrete and is identified with non-overlapping generations. Reproduction can be very generally taken as exchangeable (meaning that the labelling of individuals in each generation carries no significance). Recent generalisations have dealt with population sizes exhibiting given deterministic or (minor) random fluctuations. We consider population sizes which constitute a stationary Markov chain, explicitly allowing large fluctuations in short times. Convergence of the genealogical tree, as population size tends to infinity, towards the (time-scaled) coalescent is proved under minimal conditions. As a result, we obtain a formula for effective population size, generalising the well-known harmonic mean expression for effective size.

Keywords

Coalescentexchangeabilitypopulation genetics

MSC classification

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 92D15: Problems related to evolution 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 368 - 378
Copyright
Copyright © Applied Probability Trust 2004 

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