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Distributions of linear functionals of two parameter Poisson-Dirichlet random measures. (English) Zbl 1138.60040

Summary: The present paper provides exact expressions for the probability distributions of linear functionals of the two-parameter Poisson-Dirichlet process PD\((\alpha , \theta )\). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean of a Poisson-Dirichlet process and the mean of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are proved. The usefulness of the results contained in the paper is demonstrated by means of some illustrative examples. Indeed, our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics.

MSC:

60G57 Random measures
62F15 Bayesian inference
60E07 Infinitely divisible distributions; stable distributions
60G51 Processes with independent increments; Lévy processes
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[1] Aldous, D. and Pitman, J. (2006). Two recursive decompositions of Brownian bridge related to the asymptotics of random mappings. In Memoriam Paul-André Meyer-Séminaire de Probabilités XXXIX (M. Yor and M. Émery, eds.). Lecture Notes in Math. 1874 269-303. Springer, Berlin. · Zbl 1124.60012 · doi:10.1007/978-3-540-35513-7_19
[2] Arratia, R., Barbour, A. and Tavaré, S. (2003). Logarithmic Combinatorial Structures : A Probabilistic Approach . EMS, Zürich. · Zbl 1040.60001
[3] Barlow, M., Pitman, J. and Yor, M. (1989). Une extension multidimensionnelle de la loi de l’arc sinus. Séminaire de Probabilités XXIII (J. Azema, P.-A. Meyer and M. Yor, eds.). Lecture Notes in Math. 1372 294-314. Springer, Berlin. · Zbl 0738.60072
[4] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes . Cambridge Univ. Press. · Zbl 1107.60002
[5] Carlton, M. A. (2002). A family of densities derived from the three-parameter Dirichlet process. J. Appl. Probab. 39 764-774. · Zbl 1014.60053 · doi:10.1239/jap/1037816017
[6] Cifarelli, D. M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18 429-442. [Correction (1994) Ann. Statist. 22 1633-1634.] · Zbl 0706.62012 · doi:10.1214/aos/1176347509
[7] Cifarelli, D. M. and Mellili, E. (2000). Some new results for Dirichlet priors. Ann. Statist. 28 1390-1413. · Zbl 1105.62303 · doi:10.1214/aos/1015957399
[8] Debnath, L. (1995). Integral Transforms and Their Applications . CRC Press, Boca Raton, FL. · Zbl 0920.44001
[9] Diaconis, P. and Kemperman, J. (1996). Some new tools for Dirichlet priors. In Bayesian Statistics 5 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 97-106. Oxford Univ. Press, New York.
[10] Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230. · Zbl 0255.62037 · doi:10.1214/aos/1176342360
[11] Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615-629. · Zbl 0286.62008 · doi:10.1214/aos/1176342752
[12] Freedman, D. A. (1963). On the asymptotic behavior of Bayes’ estimates in the discrete case. Ann. Math. Statist. 34 1386-1403. · Zbl 0137.12603 · doi:10.1214/aoms/1177703871
[13] Gnedin, A. and Pitman, J. (2005). Self-similar and Markov composition structures. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov . ( POMI ) 326 , Teor. Predst. Din. Sist. Komb. i Algoritm. Metody . 13 59-84, 280-281; translation in J. Math. Sci. ( N.Y. ) 140 (2007) 376-390. · Zbl 1105.60011
[14] Gradinaru, M., Roynette, B., Vallois, P. and Yor, M. (1999). Abel transform and integrals of Bessel local times. Ann. Inst. H. Poincaré Probab. Statist. 35 531-572. · Zbl 0937.60080 · doi:10.1016/S0246-0203(99)00105-3
[15] Guglielmi, A., Holmes, C. C. and Walker, S. G. (2002). Perfect simulation involving functionals of a Dirichlet process J. Comput. Graph. Statist. 11 306-310. JSTOR: · Zbl 04576080 · doi:10.1198/106186002760180527
[16] Haas, B., Miermont, G., Pitman, J. and Winkel, M. (2006). Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Available at arxiv.org/abs/math.PR/0604350. · Zbl 1155.92033 · doi:10.1214/07-AOP377
[17] Hjort, N. L. and Ongaro, A. (2005). Exact inference for random Dirichlet means. Stat. Inference Stoch. Process. 8 227-254. · Zbl 1089.62052 · doi:10.1007/s11203-005-6068-7
[18] Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161-173. JSTOR: · Zbl 1014.62006 · doi:10.1198/016214501750332758
[19] Ishwaran, H. and James, L. F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica 13 1211-1235. · Zbl 1086.62036
[20] Ishwaran, H. and Zarepour, M. (2002). Dirichlet prior sieves in finite normal mixtures. Statist. Sinica 12 941-963. · Zbl 1002.62028
[21] James, L. F. (2002). Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics. Unpublished manuscript. Available at arxiv.org/abs/math.PR/0205093.
[22] James, L. F. (2005). Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and Beta-Gamma processes. Ann. Statist. 33 647-660. · Zbl 1071.62026 · doi:10.1214/009053604000001237
[23] James, L. F. (2006). Laws and likelihoods for Ornstein-Uhlenbeck-Gamma and other BNS OU stochastic volatility models with extensions. Available at http://arxiv.org/abs/math/0604086.
[24] Keilson, J. and Wellner, J. A. (1978). Oscillating Brownian motion. J. Appl. Probab. 15 300-310. JSTOR: · Zbl 0391.60072 · doi:10.2307/3213403
[25] Kerov, S. (1998). Interlacing measures. Amer. Math. Soc. Transl. Ser. 2 181 35-83. · Zbl 0890.05074
[26] Kerov, S. and Tsilevich, N. (1998). The Markov-Krein correspondence in several dimensions. Preprint PDMI 1/1998. · Zbl 1147.60303
[27] Lamperti, J. (1958). An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc. 88 380-387. JSTOR: · Zbl 0228.60046 · doi:10.2307/1993222
[28] Lévy, P. (1939). Sur certains processus stochastiques homogènes. Compositio Math. 7 283-339. · Zbl 0022.05903
[29] Lijoi, A. and Regazzini, E. (2004). Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 1469-1495. · Zbl 1061.60078 · doi:10.1214/009117904000000270
[30] Peccati, G. (2007). Multiple integral representation for functionals of Dirichlet processes. Bernoulli . · Zbl 1175.60072 · doi:10.3150/07-BEJ5169
[31] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21-39. · Zbl 0741.60037 · doi:10.1007/BF01205234
[32] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158. · Zbl 0821.60047 · doi:10.1007/BF01213386
[33] Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In Statistics , Probability and Game Theory (T. S. Ferguson, L. S. Shapley and J. B. MacQueen, eds.) 245-267. IMS, Hayward, CA. · doi:10.1214/lnms/1215453576
[34] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870-1902. · Zbl 0963.60079 · doi:10.1214/aop/1022677552
[35] Pitman, J. (2006). Combinatorial Stochastic Processes. Ecole d’Été de Probabilités de Saint-Flour XXXII 2002. Lecture Notes in Math. 1875 . Springer, Berlin. · Zbl 1103.60004 · doi:10.1007/b11601500
[36] Pitman, J. and Yor, M. (1992). Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. 65 326-356. · Zbl 0769.60014 · doi:10.1112/plms/s3-65.2.326
[37] Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900. · Zbl 0880.60076 · doi:10.1214/aop/1024404422
[38] Pitman, J. and Yor, M. (1997). On the relative lengths of excursions derived from a stable subordinator. Séminaire de Probabilités XXXI (J. Azema, M. Emery and M. Yor, eds.). Lecture Notes in Math. 1655 287-305. Springer, Berlin. · Zbl 0884.60072
[39] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223-252. · Zbl 0859.60067 · doi:10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-O
[40] Regazzini, E., 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. · Zbl 1018.62011 · doi:10.1214/aos/1035844980
[41] Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of random measures with independent increments. Ann. Statist. 31 560-585. · Zbl 1068.62034 · doi:10.1214/aos/1051027881
[42] Schwarz, J. H. (2005). The generalized Stieltjes transform and its inverse. J. Math. Phys. 46 013501-8. · Zbl 1076.44003 · doi:10.1063/1.1825077
[43] Sumner, D. B. (1949). An inversion formula for the generalized Stieltjes transform. Bull. Amer. Math. Soc. 55 174-183. · Zbl 0032.35501 · doi:10.1090/S0002-9904-1949-09194-2
[44] Tsilevich, N. (1997). Distribution of mean values for some random measures. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov . ( POMI ) 240 268-279. [In Russian. English translation in J. Math. Sci. 96 (1999) 3616-3623].
[45] Vershik, A., Yor, M. and Tsilevich, N. (2001). On the Markov-Krein identity and quasi-invariance of the gamma process. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov . ( POMI ) 283 21-36. [In Russian. English translation in J. Math. Sci. 121 (2004) 2303-2310]. · Zbl 1069.60046
[46] Widder, D.V. (1941). The Laplace Transform . Princeton Univ. Press. · Zbl 0063.08245
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