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On the number of allelic types for samples taken from exchangeable coalescents with mutation

Part of: Combinatorial probability Limit theorems Graph theory

Published online by Cambridge University Press:  01 July 2016

F. Freund  and
M. Möhle
F. Freund*
Affiliation:
Eberhard Karls Universität Tübingen
M. Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
*
∗ Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.
∗ Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.
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Abstract

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Let Kn denote the number of types of a sample of size n taken from an exchangeable coalescent process (Ξ-coalescent) with mutation. A distributional recursion for the sequence (Kn)n∈ℕ is derived. If the coalescent does not have proper frequencies, i.e. if the characterizing measure Ξ on the infinite simplex Δ does not have mass at 0 and satisfies ∫Δ ∣x∣Ξ(dx)/(x,x)<∞, where ∣x∣:=∑i=1∞xi and (x,x)≔∑i=1∞xi2 for x=(x1,x2,…)∈Δ, then Kn/n converges weakly as n→∞ to a limiting variable K that is characterized by an exponential integral of the subordinator associated with the coalescent process. For so-called simple measures Ξ satisfying ∫ΔΞ(d x)/(x,x)<∞, we characterize the distribution of K via a fixed-point equation.

Keywords

Coalescentdistributional recursionnumber of typessimultaneous multiple collisionsfixed pointsubordinator

MSC classification

Primary: 60C05: Combinatorial probability 05C05: Trees
Secondary: 60F05: Central limit and other weak theorems 92D15: Problems related to evolution
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 1082 - 1101
Copyright
Copyright © Applied Probability Trust 2009 

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