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

Abstract

Let K_n 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 (K_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(d x)/(x,x)<∞, where ∣x∣:=∑_i=1^∞ x_i and (x,x)≔∑_i=1^∞ x_i² for x=(x_1,x_2,...)\inΔ, then K_n/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.