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Series : Probability and Its Applications

The Poisson-Dirichlet Distribution and Related Topics, Ed. 1

Models and Asymptotic Behaviors

¥ 8,288 (tax included)

Description

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.

Contents

Part I: Models.
1. Introduction.
2. The Poisson Dirichlet Distribution.
3. The Two-Parameter Poisson Dirichlet Distribution.
4. The Coalescent.
5. Stochastic Dynamics.
6. Particle Representation.

Part II: Asymptotic Behaviors.
7. Fluctuation Theorems.
8. Large Deviations for the Poisson Dirichlet Distribution.
9. Large Deviations for the Dirichlet Processes.
A. Poisson Process and Poisson Random Measure.
.1. Definitions.
A.2. Properties.
B. Basics of Large Deviations