The infinitely-many-sites model as a measure-valued diffusion
type: Item
instance of: scholarly article
title: The infinitely-many-sites model as a measure-valued diffusion
zbMATH Open document ID: 0634.92007
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: The infinitely-many-sites model of molecular population genetics is reformulated with sites labelled by elements of [0,1] and ``type'' space \(E=[0,1]^{Z_+}\). The model is approximated by a diffusion process taking values in the set of purely atomic Borel probability measures \(\mu\) on E with the property that the locations of every \(n\geq 1\) atoms of \(\mu\) form a family tree, and the diffusion is shown to have a unique stationary distribution \({\tilde \mu}\). The principal object of investigation is the \({\tilde \mu}\)(d\(\mu)\)- expectation of the probability that a random sample from a population with types distributed according to \(\mu\) has a given tree structure. \textit{W. J. Ewens}' sampling formula [Theor. Popul. Biol. 3, 87-112 (1972; Zbl 0245.92009)] and \textit{G. A. Watterson}'s segregating-sites distribution [ibid. 7, 256-276 (1975; Zbl 0294.92011)] are obtained as corollaries.
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60K30
zbMATH DE Number: 4033557
zbMATH Keywords: diffusion approximation
zbMATH Keywords: measure-valued diffusion
zbMATH Keywords: infinitely-many-sites model
zbMATH Keywords: molecular population genetics
zbMATH Keywords: tree structure
zbMATH Keywords: sampling formula
zbMATH Keywords: segregating-sites distribution
author: Stewart N. Ethier
author: Robert Griffiths
reviewed by: Yuri Kabanov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992157
OpenAlex ID: W2026541269
DOI: 10.1214/AOP/1176992157
description: scientific article; zbMATH DE number 4033557