The Poisson-Dirichlet Distribution and Related Topics

The Poisson¿Dirichlet distribution, a probability on the in?nite-dimensional s- plex, was introduced by Kingman in 1975. Since then it has found applications in Bayesian statistics, combinatorics, number theory, ?nance, macroeconomics, physics and, especially, in population genetics. Several books have appeared that contain sections or chapters on the Poisson¿Dirichlet distribution. These include, but are not limited to, Aldous [2], Arratia, Barbour and Tavare[ ¿ 9], Ewens [67], Kingman [127, 130], and Pitman [155]. This book is the ?rst that focuses solely on the Poisson¿Dirichlet distribution and some closely related topics. The purposes of this book are to introduce the Poisson¿Dirichlet distribution, to study its connections to stochastic dynamics, and to give an up-to-date account of results concerning its various asymptotic behaviors. The book is divided into two parts. Part I, consisting of Chapters 1¿6, includes a variety of models invo- ing the Poisson¿Dirichlet distribution, and the central scheme is the uni?cation of the Poisson¿Dirichlet distribution, the urn structure, the coalescent, and the evo- tionary dynamics through the grand particle systems of Donnelly and Kurtz. Part II discusses recent progress in the study of asymptotic behaviors of the Poisson¿ Dirichlet distribution, including ?uctuation theorems and large deviations. The or- inal Poisson¿Dirichlet distribution contains one parameter denoted by?. We will also discuss an extension of this to a two-parameter distribution, where an ad- tional parameter? is needed.