The Poisson-Dirichlet Distribution and Related Topics: Models and Asymptotic Behaviors (Probability and Its Applications)

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. Front Matter....Pages I-XIII Front Matter....Pages 1-1 Introduction....Pages 3-14 The Poisson–Dirichlet Distribution....Pages 15-52 The Two-Parameter Poisson–Dirichlet Distribution....Pages 53-66 The Coalescent....Pages 67-80 Stochastic Dynamics....Pages 81-112 Particle Representation....Pages 113-124 Front Matter....Pages 125-125 Fluctuation Theorems....Pages 127-150 Large Deviations for the Poisson–Dirichlet Distribution....Pages 151-178 Large Deviations for the Dirichlet Processes....Pages 179-197 Back Matter....Pages 199-218 Pt. 1. Models. The Poisson-dirichlet Distribution -- The Two-parameter Poisson-dirichlet Distribution -- The Coalescent -- Stochastic Dynamics -- Particle Representation -- Pt 2. Asymptotic Behaviors. Fluctuation Theorems -- Large Deviations For The Poisson-dirichlet Distribution -- Large Deviations For The Dirichlet Processes -- Poisson Process And Poisson Random Measure -- Basics Of Large Deviations. Shui Feng. Includes Bibliographical References (p. 209-216) And Index. Presenting a comprehensive study of the Poisson-Dirichlet distribution, this volume emphasizes recent progress in evolutionary dynamics and asymptotic behaviors. The self-contained text presents methods and techniques that appeal to researchers in a wide variety of subjects.