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Freund, F.; Möhle, M.

On the number of allelic types for samples taken from exchangeable coalescents with mutation. (English) Zbl 1202.92061

Adv. Appl. Probab. 41, No. 4, 1082-1101 (2009).
Authors’ summary: 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 0 and satisfies \(\int _{\Delta } |x| \Xi (\text d x)/(x,x)<\infty \), where \(| x| :=\sum _{i=1}^{\infty } x_i\) and \((x,x):=\sum _{i=1}^{\infty } x_i^{2}\) for \(x=(x_1,x_2,\dots )\in\Delta \), then \(K_n/n\) converges weakly as \(n\rightarrow \infty \) 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 \(\Xi \) satisfying \(\int _{\Delta }\Xi (\text d x)/(x,x)<\infty \), we characterize the distribution of \(K\) via a fixed-point equation.
Reviewer: Claudi Alsina (Barcelona)

Cited in 7 Documents

MSC:

92D15 Problems related to evolution
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
05C05 Trees
05C90 Applications of graph theory

Keywords:

distributional recursion; number of types; simultaneous multiple collisions
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\textit{F. Freund} and \textit{M. Möhle}, Adv. Appl. Probab. 41, No. 4, 1082--1101 (2009; Zbl 1202.92061)
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