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 Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples
 Lancelot F. James, Hong Kong University of Science and TechnologyBernard Roynette, Institut Elie Cartan, Universit'{e} Henri Poincar'e,Marc Yor, Universit'e Paris VI et VII and Institut Universitaire de France

 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 $$\Gamma_{t} (G)$$ \big(resp. $$\Gamma_{t} (1/G)\big)$$ 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. AMS 2000 subject classifications: Primary 60E07, 60E10, 60G51, 60G52, 60G57.Keywords: Laplace transform, Generalized Gamma Convolutions (GGC), Wiener Gamma representation, Stieltjes transform, Dirichlet processes.
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James, Lancelot F., Roynette, Bernard, Yor, Marc, Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples, Probability Surveys, 5, (2008), 346-415 (electronic). DOI: 10.1214/07-PS118.

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