The two-parameter Poisson-Dirichlet point process

Bernoulli

Volumn 15, Issue 4, 2009, Pages 1082-1116

  Handa, Kenji ^a  

^a SAGA UNIVERSITY   (Japan)

Author keywords

Correlation function; Markov krein identity; Point process; Poisson dirichlet distribution

Indexed keywords

EID: 77649198650     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/08-BEJ180     Document Type: Article

References (61)

1. Arratia, R., Barbour, A.D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. Zurich: European Mathematical Society. MR2032426
2. Billingsley, P. (1995). Probability and Measure, 3rd ed. New York: Wiley. MR1324786
3. Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. New York: Wiley. MR1700749
4. Buchstab, A.A. (1937). An asymptotic estimation of a general number-theoretic function. Mat. Sb. 44 1239-1246.
5. Cifarelli, D.M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18 429-442;
6. correction, ibid. 22 (1994) 1633-1634. MR1041402
7. Cifarelli, D.M. and Regazzini, E. (1996). Tail behaviour and finiteness of means of distributions chosen from a Dirichlet process. Quaderno IAMI 96.19. CNR-IAMI, Milano. Available at http://www.mi.imati.cnr.it/iami/abstracts/96-19. html.
8. Daley, D.J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol.I. Elementary Theory and Methods, 2nd ed. New York: Springer. MR 1950431
9. Darling, D.A. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 95-107. MR0048726
10. de Bruijn, N.G. (1981). Asymptotic Methods in Analysis. New York: Dover Publications. (Corrected reprint of the third edition.) MR0671583
11. Derrida, B. (1997). From random walks to spin glasses. Phys. D 107 186-198. MR1491962
12. Diaconis, P. and Kemperman, J. (1996). Some new tools for Dirichlet priors. In Bayesian Statistics 5, (Alicante, 1994) 97-106. New York: Oxford Univ. Press. MR1425401
13. Dickman, K. (1930). On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astr. Fys. 22 1-14.
14. Ewens, W.J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 87-112;
15. erratum, ibid. 3 (1972) 240, 376. MR0325177
16. Feigin, P.D. and Tweedie, R.L. (1989). Linear functionals and Markov chains associated with Dirichlet processes. Math. Proc. Cambridge Philos. Soc. 105 579-585. MR0985694
17. Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol.I, 3rd ed. New York: Wiley. MR0228020
18. Feng, S. (2007). Large deviations for Dirichlet processes and Poisson-Dirichlet distribution with two parameters. Electron. J. Probab. 12 787-807. MR2318410
19. Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230. MR0350949
20. Griffiths, R.C. (1979). On the distribution of allele frequencies in a diffusion model. Theoret. Population Biol. 15 140-158. MR0528914
21. Griffiths, R.C. (1988). On the distribution of points in a Poisson Dirichlet process. J. Appl. Probab. 25 336-345. MR0938197
22. Grote, M.N. and Speed, T.P. (2002). Approximate Ewens formulae for symmetric overdominance selection. Ann. Appl. Probab. 12 637-663. MR1910643
23. Ω(n) over integers n ≤ x with all prime factors between α and y. J. Number Theory 18 206-212. MR0741951
24. Ignatov, T. (1982). On a constant arising in the asymptotic theory of symmetric groups, and Poisson-Dirichlet measures. Theory Probab. Appl. 27 136-147. MR0645134
25. James, L.F. (2005). Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and beta-gamma processes. Ann. Statist. 33 647-660. MR2163155
26. James, L.F. (2008). Large sample asymptotics for the two-parameter Poisson-Dirichlet process. IMS Collections 3 187-200. MR2459225
27. James, L.F., Lijoi, A. and Prünster, I. (2008). Distributions of linear functionals of two parameter Poisson-Dirichlet random measures. Ann. Appl. Probab. 18 521-551. MR2398765
28. James, L.F., Lijoi, A. and Prünster, I. (2009). On the posterior distribution of means of nonparametric prior processes. Bernoulli. To appear.
29. Johansson, K. (2005). Random matrices and determinantal processes. Lecture notes from the Les Houches summer school on mathematical statistical physics. Available at arXiv:math-ph/0510038.
30. [28] Joyce, P., Krone, S.M. and Kurtz, T.G. (2002). Gaussian limits associated with the Poisson-Dirichlet distribution and the Ewens sampling formula. Ann. Appl. Probab. 12 101-124. MR1890058 (Pubitemid 37023393)
31. Joyce, P., Krone, S.M. and Kurtz, T.G. (2003). When can one detect overdominant selection in the infinite-alleles model? Ann. Appl. Probab. 13 181-212. MR1951997
32. Kerov, S. (1998). Interlacing measures. In Kirillov's Seminar on Representation Theory. Amer. Math. Soc. Transi. Ser. 2 181 35-83. Providence, RI: Amer. Math. Soc. MR1618739
33. Kerov, S.V. and Tsilevich, N.V. (2001). The Markov-Krein correspondence in several dimensions. Zap. Nauchn. Sem. POMI 283 98-122;
34. translation. J. Math. Sci. (New York) 121 (2004) 2345-2359. MR1879065
35. Kingman, J.F.C. (1975). Random discrete distribution. J. Roy. Statist. Soc. Ser. B 37 1-22. MR0368264
36. Kingman, J.F.C. (1993). Poisson Processes. New York: Oxford Univ. Press. MR1207584
37. Kolchin, V.F. (1986). Random Mappings. New York: Optimization Software. MR0865130
38. Lamperti, J. (1961). A contribution to renewal theory. Proc. Amer. Math. Soc. 12 724-731. MR0125663
39. [36] Lijoi, A. and Regazzini, E. (2004). Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 1469-1495. MR2060305 (Pubitemid 38845547)
40. McCloskey, J.W. (1965). A model for the distribution of individuals by species in an environment. Ph.D. thesis, Michigan State University.
41. [38] Nieto-Barajas, L.E., Prünster, I. and Walker, S.G. (2004). Normalized random measures driven by increasing additive processes. Ann. Statist. 32 2343-2360. MR2153987 (Pubitemid 40936675)
42. Olshanski, G. (1998). Point processes and the infinite symmetric group. Part I: The general formalism and the density function. Available at arXiv:math/9804086v1. MR1671191
43. Penrose, M.D. and Wade, A.R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. in Appl. Probab. 36 691-714. MR2079909
44. Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21-39. MR1156448
45. Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158. MR1337249
46. Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. in Appl. Probab. 28 525-539. MR 1387889
47. Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875. Berlin: Springer. MR2245368
48. Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900. MR1434129
49. [46] Regazzini, E., Guglielmi, A. and Di Nunno, G. (2002). Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. Ann. Statist. 30 1376-1411. MR1936323 (Pubitemid 37095327)
50. Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist. 31 560-585. MR 1983542
51. Tennenbaum, G. (1995). Introduction to Analytic and Probabilistic Number Theory. London: Cambridge Univ. Press. MR1342300
52. Tsilevich, N. (1997). Distribution of mean values for some random measures. Zap. Nauchn. Sem. POMI 240 268-279;
53. Translation J. Math. Sci. (New York) 96 (1999) 3616-3623. MR1691650
54. Tsilevich, N., Vershik, A. and Yor, M. (2000). Distinguished properties of the gamma process, and related topics. Prépublication du Laboratoire de Probabilités et Modèles Aléatoires No.575. Available at http://xxx.lanl.gov/ps/math.PR/0005287.
55. [51] Tsilevich, N., Vershik, A. and Yor, M. (2001). An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process. J. Funct. Anal. 185 274-296. MR1853759 (Pubitemid 33119004)
56. Vershik, A. and Yor, M. (1995). Multiplicativité du processus gamma et étude asymptotique des lois stables d'indice α, lorsque α tend vers 0. Technical Report 289. Laboratoire de Probabilités, Univ. Paris VI.
57. Vershik, A., Yor, M. and Tsilevich, N. (2001). On the Markov-Krein identity and quasi-invariance of the gamma process. Zap. Nauchn. Sem. POMI 283 21-36;
58. translation. J. Math. Sci. (New York) 121 (2004) 2303-2310. MR1879060
59. Watterson, G.A. (1976). The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Probab. 13 639-651;
60. correction, ibid. 14 (1977) 897. MR0504014
61. Yamato, H. (1984). Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab. 12 262-267. MR0723745