The Poisson-Dirichlet Distribution and Related Topics ─ Models and Asymptotic Behaviors

The Poisson-Dirichlet distribution is an infinite dimensional probability distribution. It was introduced by Kingman over thirty years ago, and has found applications in a broad range of areas including Bayesian statistics, combinatorics, differential geometry, economics, number theory, physics, and population genetics. This monograph provides a comprehensive study of this distribution and some related topics, with particular emphasis on recent progresses in evolutionary dynamics and asymptotic behaviors. One central scheme is the unification of the Poisson-Dirichlet distribution, the urn structure, the coalescent, the evolutionary dynamics through the grand particle system of Donnelly and Kurtz. It is largely self-contained. The methods and techniques used in it appeal to researchers in a wide variety of subjects.

Shui Feng is a Professor of Mathematics and Statistics at McMaster University, Canada. His research is in the broad area of stochastic processes and limit theorems, with particular interests in stochastic models in population genetics, finance, and statistical physics. He is the co-editor of one book and the author of more than 30 research papers.