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Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

M. Möhle
M. Möhle*
Affiliation:
Johannes Gutenberg-Universität Mainz
*
∗ Postal address: Johannes Gutenberg-Universität Mainz, Fachbereich Mathematik, Saarstraße 21, 55099 Mainz, Germany. Email address: [email protected]
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Abstract

Haploid population models with non-overlapping generations and fixed population size N are considered. It is assumed that the family sizes ν1,…,νN within a generation are exchangeable random variables. Rates of convergence for the finite-dimensional distributions of a properly time-scaled ancestral coalescent process are established and expressed in terms of the transition probabilities of the ancestral process, i.e., in terms of the joint factorial moments of the offspring variables ν1,…,νN.

The Kingman coalescent appears in the limit as the population size N tends to infinity if and only if triple mergers are asymptotically negligible in comparison with binary mergers. In this case, a simple upper bound for the rate of convergence of the finite-dimensional distributions is derived. It depends (up to some constants) only on the three factorial moments E((ν1)2), E((ν1)3) and E((ν1)2(ν2)2), where (x)k := x(x-1)…(x-k+1). Examples are the Wright-Fisher model, where the rate of convergence is of order N-1, and the Moran model, with a convergence rate of order N-2.

Keywords

Ancestral processescoalescentexchangeabilitygeneratorneutralitypopulation geneticsrate of convergencetotal variation distanceweak convergence

MSC classification

Primary: 92D25: Population dynamics (general) 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 983 - 993
Copyright
Copyright © Applied Probability Trust 2000 

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