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Convergence to stationarity in the Moran model

Part of: Markov processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

Peter Donnelly  and
Eliane R. Rodrigues
Peter Donnelly*
Affiliation:
University of Oxford
Eliane R. Rodrigues*
Affiliation:
Instituto de Mateméticas - UNAM
*
∗Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK
∗∗Postal address: Instituto de Mateméticas - UNAM, Area de la Investigación Científica, Circuito Exterior - Ciudad Universitária, México, DF 04510, México. Email address: eliane@gauss.matem.unam.mx
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Abstract

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A1 and A2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t))t≥0 which counts the number of individuals of type A1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t∗ = Nγ-1logN + cN the separation distance between the law of X(t∗) and its stationary distribution converges to 1 - exp(-γe-γc) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.

Keywords

Rate of convergence to stationaritycut-off phenomenoncouplinggenealogical processes

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 92D10: Genetics 62E17: Approximations to distributions (nonasymptotic)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 705 - 717
Copyright
Copyright © Applied Probability Trust 2000 

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