The Annals of Probability

The Infinitely-Many-Sites Model as a Measure-Valued Diffusion

S. N. Ethier and R. C. Griffiths

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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.

Article information

Ann. Probab., Volume 15, Number 2 (1987), 515-545.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G57: Random measures
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92A10

Measure-valued diffusion population genetics infinitely-many-sites model infinitely-many-alleles model segregating sites family trees sampling distributions


Ethier, S. N.; Griffiths, R. C. The Infinitely-Many-Sites Model as a Measure-Valued Diffusion. Ann. Probab. 15 (1987), no. 2, 515--545. doi:10.1214/aop/1176992157.

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