×

The sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution. (English) Zbl 1251.60010

Pitman’s sampling formula is used to derive the Laplace transform of the two-parameter Poisson-Dirichlet distribution, and it is shown that the sampling formula uniquely determines the Laplace transform. In particular this implies that both the Laplace transform and the sampling formula characterize the two-parameter Poisson-Dirichlet distribution.
The author then derives the Laplace transform for the infinitely-many-neutral-alleles model, which is applied to prove a central limit theorem as the mutation rate converges to infinity.
Finally the Laplace transform of a non neutral model undergoing selection is derived.

MSC:

60C05 Combinatorial probability
60F05 Central limit and other weak theorems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York. · Zbl 1026.60061
[2] Ethier, S. N. (1992). Eigenstructure of the infinitely-many-neutral-alleles diffusion model. J. Appl. Prob. 29 , 487-498. · Zbl 0756.92017
[3] Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming-Viot process. Ann. Prob. 21 , 1571-1590. · Zbl 0778.60038
[4] Ethier, S. N. and Kurtz, T. G. (1981). The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13 , 429-452. · Zbl 0483.60076
[5] Feng, S. (2010). The Poisson-Dirichlet Distribution and Related Topics . Springer, Heidelberg. · Zbl 1214.60001
[6] Feng, S. and Sun, W. (2010). Some diffusion processes associated with two parameter Poisson-Dirichlet distribution and Dirichlet process. Prob. Theory Relat. Fields 148 , 501-525. · Zbl 1203.60120
[7] Feng, S., Sun, W., Wang, F. Y. and Xu, F. (2011). Functional inequalities for the two-parameter extension of the infinitely-many-neutral-alleles diffusion. J. Funct. Anal. 260 , 399-413. · Zbl 1217.60068
[8] Griffiths, R. C. (1979). A transition density expansion for a multi-allele diffusion model. Adv. Appl. Prob. 11 , 310-325. · Zbl 0405.60079
[9] Griffiths, R. C. (1979). Exact sampling distributions from the infinite neutral alleles model. Adv. Appl. Prob. 11 , 326-354. · Zbl 0406.92016
[10] Handa, K. (2005). Sampling formulae for symmetric selection. Electron. Commun. Prob. 10 , 223-234. · Zbl 1106.92055
[11] Handa, K. (2009). The two-parameter Poisson-Dirichlet point process. Bernoulli 15 , 1082-1116. · Zbl 1255.60020
[12] 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. Prob. 12 , 101-124. · Zbl 1010.62101
[13] Kingman, J. F. C. (1993). Poisson Processes . Oxford University Press. · Zbl 0771.60001
[14] Kingman, J. F. C. et al. (1975). Random discrete distributions. J. R. Statist. Soc. B 37 , 1-22. · Zbl 0331.62019
[15] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Prob. Theory Relat. Fields 92 , 21-39. · Zbl 0741.60037
[16] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Prob. Theory Relat. Fields 102 , 145-158. · Zbl 0821.60047
[17] Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In Statistics, Probability and Game Theory (IMS Lecture Notes Monogr. 30 ), Institute Mathematical Statistics, Hayward, CA, pp. 245-267.
[18] Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25 , 855-900. · Zbl 0880.60076
[19] Xu, F. (2009). A central limit theorem associated with the transformed two-parameter Poisson-Dirichlet distribution. J. Appl. Prob. 46 , 392-401. · Zbl 1173.60009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.