Hostname: page-component-7bb8b95d7b-lvwk9 Total loading time: 0 Render date: 2024-09-12T15:41:44.847Z Has data issue: false hasContentIssue false
  • English
  • Français

The sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution

Part of: Limit theorems Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Fang Xu
Fang Xu*
Affiliation:
McMaster University
*
∗ Postal address: Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we investigate the relationship between the sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution. We conclude that they are equivalent to determining the corresponding infinite-dimensional distribution. With these tools, a central limit theorem is established associated with the infinitely-many-neutral-alleles model at any fixed time. We also obtain the probability generating function of random sampling from a generalized two-parameter diffusion process. At the end of the paper a selection case is considered.

Keywords

Poisson-Dirichlet distributiontwo-parameter Poisson-Dirichlet distributionsampling formulaLaplace transform

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 1066 - 1085
Copyright
Copyright © Applied Probability Trust 2011 

References

Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Ethier, S. N. (1992). Eigenstructure of the infinitely-many-neutral-alleles diffusion model. J. Appl. Prob. 29, 487–498.Google Scholar
Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming-Viot process. Ann. Prob. 21, 1571–1590.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1981). The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452.Google Scholar
Feng, S. (2010). The Poisson–Dirichlet Distribution and Related Topics. Springer, Heidelberg.Google Scholar
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.Google Scholar
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.Google Scholar
Griffiths, R. C. (1979). A transition density expansion for a multi-allele diffusion model. Adv. Appl. Prob. 11, 310–325.Google Scholar
Griffiths, R. C. (1979). Exact sampling distributions from the infinite neutral alleles model. Adv. Appl. Prob. 11, 326–354.CrossRefGoogle Scholar
Handa, K. (2005). Sampling formulae for symmetric selection. Electron. Commun. Prob. 10, 223–234.Google Scholar
Handa, K. (2009). The two-parameter Poisson–Dirichlet point process. Bernoulli 15, 1082–1116.Google Scholar
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.Google Scholar
Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
Kingman, J. F. C. et al. (1975). Random discrete distributions. J. R. Statist. Soc. B 37, 1–22.Google Scholar
Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Prob. Theory Relat. Fields 92, 21–39.Google Scholar
Petrov, L. A. (2009). A two-parameter family of infinite-dimensional diffusions on the Kingman simplex. Funct. Anal. Appl. 43, 279–296.Google Scholar
Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Prob. Theory Relat. Fields 102, 145–158.CrossRefGoogle Scholar
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.Google Scholar
Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900.Google Scholar
Xu, F. (2009). A central limit theorem associated with the transformed two-parameter Poisson–Dirichlet distribution. J. Appl. Prob. 46, 392–401.CrossRefGoogle Scholar