The two-parameter Poisson--Dirichlet point process

TL;DR

This paper characterizes the two-parameter Poisson--Dirichlet distribution's associated point process, extending known results and exploring its mathematical structure using correlation functions, Laplace transforms, and functional analysis.

Contribution

It provides a new characterization of the two-parameter Poisson--Dirichlet point process and extends classical results from the one-parameter case using advanced mathematical tools.

Findings

Characterization of the point process via correlation functions

Extension of results from one-parameter to two-parameter case

Discussion of the Markov--Krein identity in this context

Abstract

The two-parameter Poisson--Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (that is, the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Using this, we apply the theory of point processes to reveal the mathematical structure of the two-parameter Poisson--Dirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we are able to extend several results previously known for the one-parameter case. The Markov--Krein identity for the generalized Dirichlet process is discussed from the point of view of functional analysis based on the two-parameter Poisson--Dirichlet distribution.