Abstract
The two parameter Poisson–Dirichlet distribution PD(α, θ) is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman’s Poisson–Dirichlet distribution. The two parameter Dirichlet process \({\Pi_{\alpha,\theta,\nu_0}}\) is the law of a pure atomic random measure with masses following the two parameter Poisson–Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures PD(α, θ) and \({\Pi_{\alpha,\theta,\nu_0}}\). The methods used come from the theory of Dirichlet forms.
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Bertoin J.: Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes. Combin. Prob. Comput. 17, 329–337 (2008)
Ethier S.N.: The infinitely-many-neutral-alleles diffusion model with ages. Adv. Appl. Prob. 22, 1–24 (1990)
Ethier S.N.: Eigenstructure of the infinitely-many-neutral-alleles diffusion model. J. Appl. Prob. 29, 487–498 (1992)
Ethier S.N., Kurtz T.G.: The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452 (1981)
Feng S., Wang F.Y.: A class of infinite-dimensional diffusion processes with connection to population genetics. J. Appl. Prob. 44, 938–949 (2007)
Ferguson T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)
Fukushima M., Oshima Y., Takeda M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin/New York (1994)
Griffiths R.C.: A transition density expansion for a multi-allele diffusion model. Adv. Appl. Prob. 11, 310–325 (1979)
Handa, K.: The two-parameter Poisson-Dirichlet point process. Bernoulli (to appear) (2009)
James L.F., Lijoi A., Prünster I.: Distributions of linear functionals of two parameter Poisson- Dirichlet random measures. Ann. Appl. Prob. 18, 521–551 (2008)
Kingman J.C.F.: Random discrete distributions. J. Roy. Statist. Soc. B. 37, 1–22 (1975)
Lamperti J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc. 88, 380–387 (1958)
Ma Z.M., Röckner M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer-Verlag, Berlin (1992)
Mosco U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, 368–421 (1994)
Mück S.: Large deviations w.r.t. quasi-every starting point for symmetric right processes on general state spaces. Prob. Theory Relat. Fields 99, 527–548 (1994)
Petrov, L.: A two-parameter family of infinite-dimensional diffusions in the Kingman simplex. http://arxiv.org/abs/0708.1930 (2007)
Pitman, J.: Some developments of the Blackwell-MacQueen urn scheme. Statistics, Probability, and Game Theory, pp. 245–267. IMS Lecture Notes Monogr. Ser. 30, Inst. Math. Statist., Hayward, CA (1996)
Pitman, J.: Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer (2006)
Pitman J., Yor M.: The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900 (1997)
Schmuland B.: A result on the infinitely many neutral alleles diffusion model. J. Appl. Prob. 28, 253–267 (1991)
Schmuland, B.: On the local property for positivity preserving coercive forms. Dirichlet Forms and Stochastic Processes, pp. 345–354. In: Ma, Z.M., Röckner, M., Yan, J.A. (eds.) Proceedings of the international conference held in Beijing, China, October 25–31, 1993. Walter deGruyter (1995)
Schmuland, B.: Lecture Notes on Dirichlet Forms. http://www.stat.ualberta.ca/people/schmu/preprints/yonsei.pdf (1995)
Watterson G.A.: The stationary distribution of the infinitely many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651 (1976)
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Feng, S., Sun, W. Some diffusion processes associated with two parameter Poisson–Dirichlet distribution and Dirichlet process. Probab. Theory Relat. Fields 148, 501–525 (2010). https://doi.org/10.1007/s00440-009-0238-2
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DOI: https://doi.org/10.1007/s00440-009-0238-2