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April, 1987 The Infinitely-Many-Sites Model as a Measure-Valued Diffusion
S. N. Ethier, R. C. Griffiths
Ann. Probab. 15(2): 515-545 (April, 1987). DOI: 10.1214/aop/1176992157

Abstract

The infinitely-many-sites model (with no recombination) is reformulated, with sites labelled by elements of [0, 1] and "type" space $E = \lbrack 0, 1\rbrack^{\mathbb{Z}_+}$. A gene is of type $\mathbf{x} = (x_0, x_1,\ldots) \in E$ if $x_0, x_1, \ldots$ is the sequence of sites at which mutations have occurred in the line of descent of that gene. The model is approximated by a diffusion process taking values in $\mathscr{P}^0_a(E)$, 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. Ewens' (1972) sampling formula and Watterson's (1975) segregating-sites distribution are obtained as corollaries.

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S. N. Ethier. R. C. Griffiths. "The Infinitely-Many-Sites Model as a Measure-Valued Diffusion." Ann. Probab. 15 (2) 515 - 545, April, 1987. https://doi.org/10.1214/aop/1176992157

Information

Published: April, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0634.92007
MathSciNet: MR885129
Digital Object Identifier: 10.1214/aop/1176992157

Subjects:
Primary: 60G57
Secondary: 60J70 , 92A10

Keywords: family trees , infinitely-many-alleles model , infinitely-many-sites model , measure-valued diffusion , Population genetics , sampling distributions , segregating sites

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 2 • April, 1987
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