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Convergence to the coalescent with simultaneous multiple mergers

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Serik Sagitov
Serik Sagitov*
Affiliation:
Chalmers University of Technology
*
∗Postal address: School of Mathematical Sciences, Chalmers University of Technology, S-412 96 Göteborg, Sweden. Email address: serik@math.chalmers.se
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Abstract

The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized by Möhle and Sagitov (2001) in terms of a sequence of measures defined on the finite-dimensional simplices. A more compact characterization of the general coalescent requiring a single probability measure Ξ defined on the infinite simplex Δ was suggested by Schweinsberg (2000). This paper presents a simple criterion of weak convergence to the Ξ-coalescent. In contrast to the earlier criterion of Möhle and Sagitov (2001) based on the moment conditions, the key condition here is expressed in terms of the joint distribution of the ranked offspring sizes. This criterion interprets a vector in Δ as the ranked fractions of the total population size assigned to sibling groups constituting a (rare) generation, where a merger might occur. An example of the general coalescent is developed on the basis of the Poisson–Dirichlet distribution. It suggests a simple algorithm of simulating the Kingman coalescent with occasional (simultaneous) multiple mergers.

Keywords

Ξ-coalescentexchangeabilityPoisson–Dirichlet distribution

MSC classification

Primary: 92D25: Population dynamics (general) 92D15: Problems related to evolution 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 839 - 854
Copyright
Copyright © Applied Probability Trust 2003 

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