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The infinitely-many-neutral-alleles diffusion model

Published online by Cambridge University Press:  01 July 2016

S. N. Ethier  and
Thomas G. Kurtz
S. N. Ethier*
Affiliation:
Michigan State University
Thomas G. Kurtz*
Affiliation:
University of Wisconsin-Madison
*
∗Postal address: Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, U.S.A.
∗∗Postal address: Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, U.S.A.
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Abstract

A diffusion process X(·) in the infinite-dimensional ordered simplex is characterized in terms of the generator defined on an appropriate domain. It is shown that X(·) is the limit in distribution of several sequences of discrete stochastic models of the infinitely-many-neutral-alleles type. It is further shown that X(·) has a unique stationary distribution and is reversible and ergodic. Kingman's limit theorem for the descending order statistics of the symmetric Dirichlet distribution is obtained as a corollary.

Keywords

INFINITE-DIMENSIONAL DIFFUSION PROCESSWEAK CONVERGENCESTATIONARY DISTRIBUTIONREVERSIBILITYERGODICITYPOISSON–DIRICHLET DISTRIBUTIONEWENS' SAMPLING FORMULA
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 429 - 452
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

Research supported in part by the National Science Foundation.

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