Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples

TL;DR

This paper explores properties of Generalized Gamma Convolutions (GGC), their relation to Dirichlet processes, and provides explicit examples including Bessel process excursion lengths, with a focus on Thorin measures and their representations.

Contribution

It offers new insights into GGC variables, their Thorin measures, and explicit examples, expanding understanding of their structure and relationships with Dirichlet processes.

Findings

Comparison of laws of mma_t(G) and mma_t(1/G)

Explicit examples of GGC variables including Bessel process excursions

New results on Thorin measures and their representations

Abstract

In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (:GGC) variables, their Wiener-Gamma representations, and relation with the Dirichlet processes.To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $\Gamma_t(G)$ (resp. $\Gamma_t(1/G))$ the Generalized Gamma Convolution with Thorin measure $t$-times the law of $G$ (resp. the law of $1/G$). In Section 2, we compare the laws of $\Gamma_t(G)$ and $\Gamma_t(1/G)$.In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent exponential time.