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

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 $Γ_t(G)$ (resp. $Γ_t(1/G))$ 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 $Γ_t(G)$ and $Γ_t(1/G)$.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.