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

ISBN-13: 9783642263798 / Angielski / Miękka / 2012 / 218 str.

Shui Feng

Opis

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."

Recenzje

From the reviews:

"The intended audience of this book includes researchers and graduate students in probability, population genetics and statistics. Most material in the book is self-contained and should be accessible to anyone with a graduate level knowledge of probability theory. ... this book provides useful connections to mathematical population genetics and many advanced probabilistic applications." (Wenbo V. Li, Mathematical Reviews, Issue 2012 c)

Spis treści

Models.- The Poisson–Dirichlet Distribution.- The Two-Parameter Poisson–Dirichlet Distribution.- The Coalescent.- Stochastic Dynamics.- Particle Representation.- Asymptotic Behaviors.- Fluctuation Theorems.- Large Deviations for the Poisson–Dirichlet Distribution.- Large Deviations for the Dirichlet Processes.

O autorze

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.

Blurb

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.