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

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

2008-08-13

arxiv.org/abs/0808.1792v1

Mathematics

Probability

21 pages

Scientific paper

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.

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