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

Abstract

LetKndenote the number of types of a sample of sizentaken from an exchangeable coalescent process (Ξ-coalescent) with mutation. A distributional recursion for the sequence (Kn)n∈ℕis 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∞xiand (x,x)≔∑i=1∞xi2forx=(x1,x2,…)∈Δ, thenKn/nconverges weakly asn→∞ to a limiting variableKthat is characterized by an exponential integral of the subordinator associated with the coalescent process. For so-called simple measures Ξ satisfying ∫ΔΞ(dx)/(x,x)<∞, we characterize the distribution ofKvia a fixed-point equation.