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Convergence to stationarity in the Moran model

  • Peter Donnelly (a1) and Eliane R. Rodrigues (a2)

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 A 1 and A 2. 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 A 1. 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(-γec ) 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.

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Corresponding author

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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