The infinitely-many-sites model as a measure-valued diffusion
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.
scientific article; zbMATH DE number 4033557