## Probability Surveys

### 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.

#### Article information

Source
Probab. Surveys, Volume 5 (2008), 346-415.

Dates
First available in Project Euclid: 10 October 2008

https://projecteuclid.org/euclid.ps/1223654264

Digital Object Identifier
doi:10.1214/07-PS118

Mathematical Reviews number (MathSciNet)
MR2476736

Zentralblatt MATH identifier
1189.60035

#### Citation

James, Lancelot F.; Roynette, Bernard; Yor, Marc. Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surveys 5 (2008), 346--415. doi:10.1214/07-PS118. https://projecteuclid.org/euclid.ps/1223654264

#### References

• [1] Barndorff-Nielsen, O. E., Maejima, M. and Sato, K.I. (2006). Some classes of multivariate infinitely divisible distribution admitting stochastic integral representations Bernoulli, 12, p. 1-33.
• [2] Bertoin, J. (1996). Lévy processes. Cambridge Tracts in Mathematics, 121, Cambridge University Press.
• [3] Bertoin, J. (2000). The convex minorant of the Cauchy process. Electron. Comm. Probab. 5 51-55.
• [4] Bertoin, J., Fujita, T., Roynette, B. and Yor, M. (2006). On a particular class of self-decomposable random variables: the duration of a Bessel excursion straddling an independent exponential time. Prob. Math. Stat. 26, 315-366.
• [5] Bondesson, L. (1992). Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics, 76. Springer-Verlag, New York.
• [6] Bondesson, L. (1981). Classes of infinitely divisible distributions and densities. Z. Wahrsch. Verw. Gebiete 57 39-71.
• [7] Bondesson, L. (1990). Generalized gamma convolutions and complete monotonicity. Probab. Theory Related Fields 85 181–194.
• [8] Chaumont, L. and Yor, M. (2003). Exercises in probability. A guided tour from measure theory to random processes, via conditioning. Cambridge Series in Statistical and Probabilistic Mathematics, 13, Cambridge University Press.
• [9] Cifarelli, D.M. and Mellili, E. (2000). Some new results for Dirichlet priors. Ann. Statist. 28, 1390–1413.
• [10] Cifarelli, D. M. and Regazzini, E. (1979). Considerazioni generali sull’impostazione bayesiana di problemi non parametrici. Le medie associative nel contesto del processo aleatorio di Dirichlet I, II. Riv. Mat. Sci. Econom. Social 2, 39–52.
• [11] Cifarelli, D.M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18, 429–442 (Correction in Ann. Statist. (1994) 22, 1633–1634).
• [12] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev., 41, 45-76.
• [13] Diaconis, P. and Kemperman, J. (1996). Some new tools for Dirichlet priors. In Bayesian Statistics 5 (Bernardo, J.M., Berger, J.O., Dawid, A.P. and Smith, A.F.M., Eds.), 97–106. Oxford University Press, New York.
• [14] Doss, H. and Sellke, T. (1982). The tails of probabilities chosen from a Dirichlet prior. Ann. Statist. 10, 1302–1305.
• [15] Dufresne, D. and Yor, M. (2007). In preparation.
• [16] Epifani, I., Guglielmi, A. and Melilli, E. (2004). Some new results on random Dirichlet variances. Technical Report IMATI 2004-15-MI (2004).
• [17] Epifani, I., Guglielmi, A. and Melilli, E. (2006). A stochastic equation for the law of the random Dirichlet variance. Statist. Probab. Lett. 76 495–502.
• [18] Emery, M. and Yor, M. (2004). A parallel between Brownian bridges and gamma bridges. Publ. Res. Inst. Math. Sci., Kyoto University, 40 669-688.
• [19] Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming-Viot process Ann. Probab. 21 1571–1590.
• [20] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230.
• [21] Feigin, P.D. and Tweedie, R.L. (1989). Linear functionals and Markov chains associated with Dirichlet processes. Math. Proc. Philos. Soc. 105, p. 579-585.
• [22] Freedman, D.A. On the asymptotic behavior of Bayes’ estimates in the discrete case. (1963). Ann. Math. Statist., 34, 1386-1403.
• [23] Fujita. T. and Yor, M. (2006). An interpretation of the results of the BFRY paper in terms of certain means of Dirichlet Processes.
• [24] Goldie, C. A class of infinitely divisible random variables. (1967). Proc. Cambridge Philos. Soc. 63 1141-1143.
• [25] Grigelionis, B. (2007). Extended Thorin classes and stochastic integrals. Lithuanian Math. Journal 47, n 4, p. 406-411.
• [26] Hannum, R.C., Hollander, M. and Langberg, N.A. (1981). Distributional results for random functionals of a Dirichlet process Ann. Probab. 9 665-670.
• [27] Hjort, N.L. and Ongaro, A. (2005). Exact inference for random Dirichlet means. Stat. Inference Stoch. Process. 8 227-254.
• [28] James, L.F. (2006). Gamma tilting calculus for GGC and Dirichlet means via applications to Linnik processes and occupation time laws for randomly skewed Bessel processes and bridges. http://arxiv.org/abs/math.PR/0610218.
• [29] James, L.F. (2007). New Dirichlet mean identities. http://arxiv.org/abs/0708.0614.
• [30] James, L. F., Lijoi, A. and Prünster, I. (2008). Distributions of functionals of the two parameter Poisson-Dirichlet process. Ann. Appl. Probab. 18, 521–551.
• [31] James, L.F. and Yor, M. (2006). Tilted stable subordinators, Gamma time changes and Occupation Time of rays by Bessel Spiders. http://arxiv.org/abs/math.PR/0701049.
• [32] Jeanblanc, M., Pitman, J. and Yor, M. (2002). Self-similar processes with independent increments associated with Lévy and Bessel processes. Stochastic Process. Appl. 100 223–231.
• [33] Lamperti, J. (1958). An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc., 88, p. 380-387 (1958).
• [34] Lebedev, N.N. (1972). Special functions and their applications. Revised edition, translated from the Russian and edited by Richard A. Silverman.
• [35] Letac, G. (1985). A characterization of the Gamma distribution. Adv. in Appl. Probab. 17 911-912.
• [36] Lijoi, A. and Regazzini, E. (2004). Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 1469-1495.
• [37] Lukacs, E. (1970). Characteristic functions. Second edition, revised and enlarged. Hafner Publishing Co., New York.
• [38] Matsumoto, H., Nguyen, L. and Yor, M. (2002). Subordinators related to the exponential functionals of Brownian bridges and explicit formulae for the semigroups of hyperbolic Brownian motions. Stochastic processes and related topics (Siegmundsburg, 2000), p. 213-235, Stochastic Monogr., 12, Taylor and Francis, London.
• [39] Möhle, M. (2005). Convergence results for compound Poisson distributions and applications to the standard Luria-Delbruck distribution. J.Applied Prob. 42 620-631.
• [40] Patterson, S.J. (1998). An introduction to the theory of the Riemann zeta-function. Cambridge Studies in Advanced Mathematics, 14, Cambridge University Press.
• [41] Pitman, J. and Yor, M. (2003). Infinitely divisible laws associated with hyperbolic functions. Canad. J. Math. 55 292-330.
• [42] Regazzini, E., A. Guglielmi, A. and Di Nunno, G. (2002). Theory and numerical analysis for exact distribution of functionals of a Dirichlet process. Ann. Statist. 30, 1376-1411.
• [43] Revuz, D. and Yor, M. (1999). Continuous martingales and Brownian motion. Third edition, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 293, Springer-Verlag, Berlin.
• [44] Roynette, B. and Yor, M. (2005). Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d’Euler et à la fonction zêta de Riemann. (Infinitely divisible Wald pairs: examples associated with the Euler gamma function and the Riemann zeta function.) Ann. Inst. Fourier (Grenoble) 55 1219-1283.
• [45] Sato, K. I. (1999). Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press.
• [46] Song, R. and Vondracek, Z. (2006). Potential theory of special subordinators and subordinate killed stable processes. J. Theor. Prob. 19 807-847.
• [47] Steutel, F. W. (1967). Note on the infinite divisibility of exponential mixtures. Ann. Math. Statist. 38 1303-1305.
• [48] Steutel, F. W. and Van Harn, K. (2004). Infinite divisibility of probability distributions on the real line. Monographs and Textbooks in Pure and Applied Mathematics, 259. Marcel Dekker.
• [49] Thorin, O. (1977). On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 3 121-148.
• [50] Vershik, A., Yor, M. and Tsilevich, N. (2004). On the Markov–Krein identity and quasi–invariance of the gamma process. J. Math. Sci. 121 2303-2310.
• [51] Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. in Appl. Probab. 11, 780-783.
• [52] Watson, G. N. (1944). A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, England; The Macmillan Company, New York.
• [53] Widder, D.V. (1941). The Laplace Transform. Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J..
• [54] Winkel, M. (2005). Electronic foreign-exchange markets and passage events of independent subordinators. J. Appl. Probab. 42, 138-152.
• [55] Yor, M. (2007). Some remarkable properties of the Gamma process. Festschrift for Dilip Madan, Advances in Mathematical Finance. p. 37-47, Applied Numer. Harmon And., eds M. Fu, R. Jarrow, R. Elliott, Birkhäuser Boston.
• [56] Yamato, H. (1984). Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab. 12 262-267.