2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/166972Let $K_n$ 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 $(K_n)_{n\in{\mathbb N}}$ 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 zero and satisfies $\int_Δ|x|Ξ(dx)/(x,x)<\infty$, where $|x|:=\sum_{i=1}^\infty x_i$ and $(x,x):=\sum_{i=1}^\infty x_i^2$ for $x=(x_1,x_2,...)\inΔ$, then $K_n/n$ converges weakly as $n\to\infty$ to a limiting variable $K$ which is characterized by an exponential integral of the subordinator associated with the coalescent process. For so-called simple measures $Ξ$ satisfying $\int_ΔΞ(dx)/(x,x)<\infty$ we characterize the distribution of $K$ via a fixed-point equation.21 pagesProbability60C05; 05C05; 60F05; 92D15text