Gamma-Dirichlet Structure and Two Classes of Measure-valued Processes

TL;DR

This paper explores the Gamma-Dirichlet structure linking gamma and Dirichlet processes, introduces new results on large deviations and quasi-invariance, and examines their roles as reversible measures in measure-valued processes.

Contribution

It provides new insights into the Gamma-Dirichlet structure, including large deviations, quasi-invariance, and the dynamical relation between measure-valued processes.

Findings

Derived the transition function of the Fleming-Viot process from the branching diffusion.

Established the reversibility of the measure-valued branching diffusion.

Provided new results on large deviations and quasi-invariance of the gamma process.

Abstract

The Gamma-Dirichlet structure corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief review of existing results concerning the Gamma-Dirichlet structure. New results are obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. The laws of the gamma process and the Dirichlet process are the respective reversible measures of the measure-valued branching diffusion with immigration and the Fleming-Viot process with parent independent mutation. We view the relation between these two classes of measure-valued processes as the dynamical Gamma-Dirichlet structure. Other results of this article include the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration, and the establishment of the reversibility of the latter. One of these is related to an open problem by Ethier and Griffiths and the other leads to an alternative proof of the reversibility of the Fleming-Viot process.