Generalized gamma convolutions, dirichlet means, thorin measures, with explicit examples

Probability Surveys

Volumn 5, Issue 1, 2008, Pages 346-415

  James, Lancelot F ^a   Roynette, Bernard ^b   Yor, Marc ^c  

^a HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Hong Kong)

^b UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

^c SORBONNE UNIVERSITÉ   (France)

Author keywords

Dirich let processes; Generalized Gamma Convolutions (GGC); Laplace transform; Stieltjes transform; Wiener Gamma representation

Indexed keywords

EID: 68749088428     PISSN: None     EISSN: 15495787     Source Type: Journal    
DOI: 10.1214/07-PS118     Document Type: Article

References (56)

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. MR2202318
2. Bertoin, J. (1996). Lévy processes. Cambridge Tracts in Mathematics, 121, Cambridge University Press. MR1406564
3. Bertoin, J. (2000). The convex minorant of the Cauchy process. Electron. Comm. Probab. 5 51-55. 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. MR2325310
5. Bondesson, L. (1992). Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics, 76. Springer-Verlag, New York. MR1224674
6. Bondesson, L. (1981). Classes of infinitely divisible distributions and densities. Z. Wahrsch. Verw. Gebiete 57 39-71. MR0623454
7. Bondesson, L. (1990). Generalized gamma convolutions and complete monotonicity. Probab. Theory Related Fields 85 181-194. MR1050743
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. MR2016344
9. Cifarelli, D.M. and Mellili, E. (2000). Some new results for Dirichlet priors. Ann. Statist. 28, 1390-1413. MR1805789
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. MR0573686
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). MR1041402
12. Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev., 41, 45-76. MR1669737
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. MR1425401
14. Doss, H. and Sellke, T. (1982). The tails of probabilities chosen from a Dirichlet prior. Ann. Statist. 10, 1302-1305. MR0673666
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. 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. MR2074696
19. Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming-Viot process Ann. Probab. 21 1571-1590. MR1235429
20. Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230. MR0350949
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. MR0985694
22. Freedman, D.A. On the asymptotic behavior of Bayes' estimates in the discrete case. (1963). Ann. Math. Statist., 34, 1386-1403. MR0158483
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. MR0215332
25. Grigelionis, B. (2007). Extended Thorin classes and stochastic integrals. Lithuanian Math. Journal 47, n? 4, p. 406-411. 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. MR0630318
27. Hjort, N.L. and Ongaro, A. (2005). Exact inference for random Dirichlet means. Stat. Inference Stoch. Process. 8 227-254. MR2177313
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. MR2275248
29. James, L.F. (2007). New Dirichlet mean identities. http://arxiv.org/abs/0708.0614.
30. James, L. F., Lijoi, A. and Pr̈unster, I. (2008). Distributions of functionals of the two parameter Poisson-Dirichlet process. Ann. Appl. Probab. 18, 521-551. MR2398765
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. MR1919614
33. Lamperti, J. (1958). An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc., 88, p. 380-387 (1958). MR0094863
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. MR0809436
36. Lijoi, A. and Regazzini, E. (2004).Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 1469-1495. MR2060305
37. Lukacs, E. (1970). Characteristic functions. Second edition, revised and enlarged. Hafner Publishing Co., New York. MR0346874
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. MR1987318
39. M̈ohle, M. (2005). Convergence results for compound Poisson distributions and applications to the standard Luria-Delbruck distribution. J.Applied Prob. 42 620-631. MR2157509
40. Patterson, S.J. (1998). An introduction to the theory of the Riemann zeta-function. Cambridge Studies in Advanced Mathematics, 14, Cambridge University Press. MR0933558
41. Pitman, J. and Yor, M. (2003). Infinitely divisible laws associated with hyperbolic functions. Canad. J. Math. 55 292-330. 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. MR1936323
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. MR1725357
44. Roynette, B. and Yor, M. (2005). Couples de Wald indéfiniment divisibles. Exemples liés a la fonction gamma d'Euler et a la fonction ẑeta 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. MR2157168
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. MR1739520
46. Song, R. and Vondracek, Z. (2006). Potential theory of special subordinators and subordinate killed stable processes. J. Theor. Prob. 19 807-847. MR2279605
47. Steutel, F. W. (1967). Note on the infinite divisibility of exponential mixtures. Ann. Math. Statist. 38 1303-1305. MR0215339
48. Steutel, F. W. and Van Harn, K. (2004). Infinite divisibility of prob- ability distributions on the real line. Monographs and Textbooks in Pure and Applied Mathematics, 259. Marcel Dekker. MR2011862
49. Thorin, O. (1977). On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 3 121-148. MR0552135
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. 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. MR0544194
52. Watson, G. N. (1944). A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, England; The Macmillan Company, New York. MR0010746
53. Widder, D.V. (1941). The Laplace Transform. Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J. MR0005923
54. Winkel, M. (2005). Electronic foreign-exchange markets and passage events of independent subordinators. J. Appl. Probab. 42, 138-152. MR2144899
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, Birkḧauser Boston. MR2359361
56. Yamato, H. (1984). Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab. 12 262-267. MR0723745