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On the number of segregating sites for populations with large family sizes

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

M Möhle
M Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
*
∗ Postal address: Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: martin.moehle@uni-tuebingen.de
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Abstract

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We present recursions for the total number, Sn, of mutations in a sample of n individuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rate r>0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of Λ-coalescent processes allowing for multiple collisions, such that the measure Λ(dx)/x is finite, we prove that Sn/(nr) converges in distribution to a limiting variable, S, characterized via an exponential integral of a certain subordinator. When the measure Λ(dx)/x2 is finite, the distribution of S coincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the form with specific independent random coefficients A and B. Examples are presented in which explicit representations for (the density of) S are available. We conjecture that Sn/E(Sn)→1 in probability if the measure Λ(dx)/x is infinite.

Keywords

Autoregressive processcoalescent processinfinitely-many-sites modelmultiple collisionssegregating sitesstochastic difference equationtotal number of mutations

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 92D10: Genetics
Secondary: 60F05: Central limit and other weak theorems 92D15: Problems related to evolution
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 750 - 767
Copyright
Copyright © Applied Probability Trust 2006 

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