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 ${\Gamma }_t(G)$ (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.

Published in at http://dx.doi.org/10.1214/07-PS118 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Means of a Dirichlet process and multiple hypergeometric functions
• Distributions of linear functionals of two parameter Poisson--Dirichlet random measures

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