Advances in Applied Probability

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


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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∈∕ 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 ∫Δx∣Ξ(d x)/(x,x)<∞, where ∣x∣:=∑i=1 xi and (x,x)≔∑i=1 xi2 for x=(x_1,x_2,...)\inΔ, 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.

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Adv. in Appl. Probab. Volume 41, Number 4 (2009), 1082-1101.

First available in Project Euclid: 24 December 2009

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05C05: Trees
Secondary: 60F05: Central limit and other weak theorems 92D15: Problems related to evolution

Coalescent distributional recursion number of types simultaneous multiple collisions fixed point subordinator


FREUND, F.; MÖHLE, M. On the number of allelic types for samples taken from exchangeable coalescents with mutation. Adv. in Appl. Probab. 41 (2009), no. 4, 1082--1101. doi:10.1239/aap/1261669587.

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