## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 2 (1987), 431-835

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

S. N. Ethier and R. C. Griffiths

#### 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.

#### Article information

**Source**

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

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176992157

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176992157

**Mathematical Reviews number (MathSciNet)**

MR885129

**Zentralblatt MATH identifier**

0634.92007

**Subjects**

Primary: 60G57: Random measures

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

**Keywords**

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

#### Citation

Ethier, S. N.; Griffiths, R. C. The Infinitely-Many-Sites Model as a Measure-Valued Diffusion. The Annals of Probability 15 (1987), no. 2, 515--545. doi:10.1214/aop/1176992157. http://projecteuclid.org/euclid.aop/1176992157.