## Advances in Applied Probability

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

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 24 December 2009

Permanent link to this document
https://projecteuclid.org/euclid.aap/1261669587

Digital Object Identifier
doi:10.1239/aap/1261669587

Mathematical Reviews number (MathSciNet)
MR2663237

Zentralblatt MATH identifier
1202.92061

#### Citation

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. https://projecteuclid.org/euclid.aap/1261669587.

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