Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples
Copyright policy )Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples
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 $��_t(G)$ (resp. $��_t(1/G))$ 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 $��_t(G)$ and $��_t(1/G)$.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)
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Dirichlet processes, Wiener gamma representation, Dirich-let processes, Laplace transform, Infinitely divisible distributions; stable distributions, Processes with independent increments; Lévy processes, Research exposition (monographs, survey articles) pertaining to probability theory, Characteristic functions; other transforms, 60E07, FOS: Mathematics, Generalized Gamma Convolutions (GGC), Probability (math.PR), Wiener Gamma representation, generalized gamma convolutions (GGC), Stable stochastic processes, Stieltjes transform, 60G57, 60E10, 60G51, Mathematics - Probability, 60G52, Random measures, 60E07, 60E10, 60G51, 60G52, 60G57 (Primary)
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Dirichlet processes, Wiener gamma representation, Dirich-let processes, Laplace transform, Infinitely divisible distributions; stable distributions, Processes with independent increments; Lévy processes, Research exposition (monographs, survey articles) pertaining to probability theory, Characteristic functions; other transforms, 60E07, FOS: Mathematics, Generalized Gamma Convolutions (GGC), Probability (math.PR), Wiener Gamma representation, generalized gamma convolutions (GGC), Stable stochastic processes, Stieltjes transform, 60G57, 60E10, 60G51, Mathematics - Probability, 60G52, Random measures, 60E07, 60E10, 60G51, 60G52, 60G57 (Primary)
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