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

TL;DR

This paper derives a recursion for the number of allelic types in samples from exchangeable coalescents with mutation, and characterizes the limiting distribution of scaled types under certain conditions.

Contribution

It introduces a distributional recursion for $K_n$ and characterizes the limit distribution of $K_n/n$ for a broad class of $ ext{Xi}$-coalescents with mutation.

Findings

$K_n/n$ converges weakly to a limiting variable $K$ under specified conditions.

The distribution of $K$ is characterized by an exponential integral of an associated subordinator.

For simple measures $ ext{Xi}$, the distribution of $K$ satisfies a fixed-point equation.

Abstract

Let $K_n$ denote the number of types of a sample of size $n$ taken from an exchangeable coalescent process ($\Xi$-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 $\Xi$ on the infinite simplex $\Delta$ does not have mass at zero and satisfies $\int_\Delta |x|\Xi(dx)/(x,x)<\infty$, where $|x|:=\sum_{i=1}^\infty x_i$ and $(x,x):=\sum_{i=1}^\infty x_i^2$ for $x=(x_1,x_2,...)\in\Delta$, 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 $\Xi$ satisfying $\int_\Delta\Xi(dx)/(x,x)<\infty$ we characterize the distribution of $K$ via a fixed-point equation.