Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples
Lancelot F. James, Bernard Roynette, and Marc Yor
Source: Probab. Surveys Volume 5 (2008), 346-415.
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 Γt(G) (resp. Γt(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 Γt(G) and Γt(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.
Primary Subjects: 60E07, 60E10, 60G51, 60G52, 60G57
Keywords: Laplace transform; Generalized Gamma Convolutions (GGC); Wiener Gamma representation; Stieltjes transform; Dirichlet processes
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ps/1223654264
Digital Object Identifier: doi:10.1214/07-PS118
Mathematical Reviews number (MathSciNet):
MR2476736
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.
Mathematical Reviews (MathSciNet):
MR2202318
Project Euclid: euclid.bj/1141136646
[2] Bertoin, J. (1996). Lévy processes. Cambridge Tracts in Mathematics, 121, Cambridge University Press.
Mathematical Reviews (MathSciNet):
MR1406564
[3] Bertoin, J. (2000). The convex minorant of the Cauchy process. Electron. Comm. Probab. 5 51-55.
Mathematical Reviews (MathSciNet):
MR1747095
[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.
Mathematical Reviews (MathSciNet):
MR2325310
[5] Bondesson, L. (1992). Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics, 76. Springer-Verlag, New York.
Mathematical Reviews (MathSciNet):
MR1224674
[6] Bondesson, L. (1981). Classes of infinitely divisible distributions and densities. Z. Wahrsch. Verw. Gebiete 57 39-71.
Mathematical Reviews (MathSciNet):
MR623454
[7] Bondesson, L. (1990). Generalized gamma convolutions and complete monotonicity. Probab. Theory Related Fields 85 181–194.
Mathematical Reviews (MathSciNet):
MR1050743
Digital Object Identifier: doi:10.1007/BF01277981
[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.
Mathematical Reviews (MathSciNet):
MR2016344
[9] Cifarelli, D.M. and Mellili, E. (2000). Some new results for Dirichlet priors. Ann. Statist. 28, 1390–1413.
Mathematical Reviews (MathSciNet):
MR1805789
Digital Object Identifier: doi:10.1214/aos/1015957399
Project Euclid: euclid.aos/1015957399
[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.
Mathematical Reviews (MathSciNet):
MR574040
Digital Object Identifier: doi:10.1007/BF02620300
[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).
Mathematical Reviews (MathSciNet):
MR1041402
Digital Object Identifier: doi:10.1214/aos/1176347509
Project Euclid: euclid.aos/1176347509
[12] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev., 41, 45-76.
Mathematical Reviews (MathSciNet):
MR1669737
Digital Object Identifier: doi:10.1137/S0036144598338446
JSTOR: links.jstor.org
[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.
Mathematical Reviews (MathSciNet):
MR1425401
[14] Doss, H. and Sellke, T. (1982). The tails of probabilities chosen from a Dirichlet prior. Ann. Statist. 10, 1302–1305.
Mathematical Reviews (MathSciNet):
MR673666
Digital Object Identifier: doi:10.1214/aos/1176345996
Project Euclid: euclid.aos/1176345996
[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.
Mathematical Reviews (MathSciNet):
MR2266603
[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.
Mathematical Reviews (MathSciNet):
MR2074696
Digital Object Identifier: doi:10.2977/prims/1145475488
Project Euclid: euclid.prims/1145475488
[19] Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming-Viot process Ann. Probab. 21 1571–1590.
Mathematical Reviews (MathSciNet):
MR1235429
Digital Object Identifier: doi:10.1214/aop/1176989131
Project Euclid: euclid.aop/1176989131
[20] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230.
Mathematical Reviews (MathSciNet):
MR350949
Digital Object Identifier: doi:10.1214/aos/1176342360
Project Euclid: euclid.aos/1176342360
[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.
Mathematical Reviews (MathSciNet):
MR985694
[22] Freedman, D.A. On the asymptotic behavior of Bayes’ estimates in the discrete case. (1963). Ann. Math. Statist., 34, 1386-1403.
Mathematical Reviews (MathSciNet):
MR158483
Digital Object Identifier: doi:10.1214/aoms/1177703871
Project Euclid: euclid.aoms/1177703871
[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.
Mathematical Reviews (MathSciNet):
MR215332
Digital Object Identifier: doi:10.1017/S0305004100042225
[25] Grigelionis, B. (2007). Extended Thorin classes and stochastic integrals. Lithuanian Math. Journal 47, n○ 4, p. 406-411.
Mathematical Reviews (MathSciNet):
MR2392715
[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.
Mathematical Reviews (MathSciNet):
MR630318
Digital Object Identifier: doi:10.1214/aop/1176994373
Project Euclid: euclid.aop/1176994373
[27] Hjort, N.L. and Ongaro, A. (2005). Exact inference for random Dirichlet means. Stat. Inference Stoch. Process. 8 227-254.
Mathematical Reviews (MathSciNet):
MR2177313
Digital Object Identifier: doi:10.1007/s11203-005-6068-7
[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.
Mathematical Reviews (MathSciNet):
MR2222511
[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.
Mathematical Reviews (MathSciNet):
MR1919614
Digital Object Identifier: doi:10.1016/S0304-4149(02)00098-4
[33] Lamperti, J. (1958). An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc., 88, p. 380-387 (1958).
Mathematical Reviews (MathSciNet):
MR94863
Digital Object Identifier: doi:10.2307/1993222
JSTOR: links.jstor.org
[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.
Mathematical Reviews (MathSciNet):
MR809436
Digital Object Identifier: doi:10.2307/1427095
JSTOR: links.jstor.org
[36] Lijoi, A. and Regazzini, E. (2004). Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 1469-1495.
Mathematical Reviews (MathSciNet):
MR2060305
Digital Object Identifier: doi:10.1214/009117904000000270
Project Euclid: euclid.aop/1084884858
[37] Lukacs, E. (1970). Characteristic functions. Second edition, revised and enlarged. Hafner Publishing Co., New York.
Mathematical Reviews (MathSciNet):
MR346874
[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.
Mathematical Reviews (MathSciNet):
MR1987318
[39] Möhle, M. (2005). Convergence results for compound Poisson distributions and applications to the standard Luria-Delbruck distribution. J.Applied Prob. 42 620-631.
Mathematical Reviews (MathSciNet):
MR2157509
Digital Object Identifier: doi:10.1239/jap/1127322016
Project Euclid: euclid.jap/1127322016
[40] Patterson, S.J. (1998). An introduction to the theory of the Riemann zeta-function. Cambridge Studies in Advanced Mathematics, 14, Cambridge University Press.
Mathematical Reviews (MathSciNet):
MR933558
[41] Pitman, J. and Yor, M. (2003). Infinitely divisible laws associated with hyperbolic functions. Canad. J. Math. 55 292-330.
Mathematical Reviews (MathSciNet):
MR1969794
[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.
Mathematical Reviews (MathSciNet):
MR1936323
Digital Object Identifier: doi:10.1214/aos/1035844980
Project Euclid: euclid.aos/1035844980
[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.
Mathematical Reviews (MathSciNet):
MR1725357
[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.
Mathematical Reviews (MathSciNet):
MR1739520
[46] Song, R. and Vondracek, Z. (2006). Potential theory of special subordinators and subordinate killed stable processes. J. Theor. Prob. 19 807-847.
Mathematical Reviews (MathSciNet):
MR2279605
Digital Object Identifier: doi:10.1007/s10959-006-0045-y
[47] Steutel, F. W. (1967). Note on the infinite divisibility of exponential mixtures. Ann. Math. Statist. 38 1303-1305.
Mathematical Reviews (MathSciNet):
MR215339
Digital Object Identifier: doi:10.1214/aoms/1177698806
Project Euclid: euclid.aoms/1177698806
[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.
Mathematical Reviews (MathSciNet):
MR2011862
[49] Thorin, O. (1977). On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 3 121-148.
Mathematical Reviews (MathSciNet):
MR552135
[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.
Mathematical Reviews (MathSciNet):
MR1879060
[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.
Mathematical Reviews (MathSciNet):
MR544194
Digital Object Identifier: doi:10.2307/1426858
JSTOR: links.jstor.org
[52] Watson, G. N. (1944). A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, England; The Macmillan Company, New York.
Mathematical Reviews (MathSciNet):
MR10746
[53] Widder, D.V. (1941). The Laplace Transform. Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J..
Mathematical Reviews (MathSciNet):
MR5923
[54] Winkel, M. (2005). Electronic foreign-exchange markets and passage events of independent subordinators. J. Appl. Probab. 42, 138-152.
Mathematical Reviews (MathSciNet):
MR2144899
Digital Object Identifier: doi:10.1239/jap/1110381376
Project Euclid: euclid.jap/1110381376
[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.
Mathematical Reviews (MathSciNet):
MR2359361
Digital Object Identifier: doi:10.1007/978-0-8176-4545-8_3
[56] Yamato, H. (1984). Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab. 12 262-267.
Mathematical Reviews (MathSciNet):
MR723745
Digital Object Identifier: doi:10.1214/aop/1176993389
Project Euclid: euclid.aop/1176993389
