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 Probability Surveys > Vol. 5 (2008) open journal systems 

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

Lancelot F. James, Hong Kong University of Science and Technology
Bernard Roynette, Institut Elie Cartan, Universit'{e} Henri Poincar'e,
Marc Yor, Universit'e Paris VI et VII and Institut Universitaire de France

  • 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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