Distinguished properties of the gamma process and related topics

TL;DR

This paper explores the fundamental properties of the gamma process, revealing its invariance, its relation to stable processes, and its connections to Poisson-Dirichlet measures, highlighting its deep similarities with Brownian motion.

Contribution

It establishes the quasi-invariance of the gamma process, its limit relation to stable processes, and its invariance properties, providing new insights into its connection with Poisson-Dirichlet measures.

Findings

Gamma process is quasi-invariant under large linear groups

It is a renormalized limit of stable processes

Gamma process has an invariant sigma-finite measure

Abstract

We study fundamental properties of the gamma process and their relation to various topics such as Poisson-Dirichlet measures and stable processes. We prove the quasi-invariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit of the stable processes and has an equivalent sigma-finite measure (quasi-Lebesgue) with important invariance properties. New properties of the gamma process can be applied to the Poisson-Dirichlet measures. We also emphasize the deep similarity between the gamma process and the Brownian motion. The connection of the above topics makes more transparent some old and new facts about stable and gamma processes, and the Poisson-Dirichlet measures.