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Some diffusion processes associated with two parameter Poisson–Dirichlet distribution and Dirichlet process

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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References

  1. 1

    Bertoin J.: Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes. Combin. Prob. Comput. 17, 329–337 (2008)

    MATH  MathSciNet  Google Scholar 

  2. 2

    Ethier S.N.: The infinitely-many-neutral-alleles diffusion model with ages. Adv. Appl. Prob. 22, 1–24 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3

    Ethier S.N.: Eigenstructure of the infinitely-many-neutral-alleles diffusion model. J. Appl. Prob. 29, 487–498 (1992)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4

    Ethier S.N., Kurtz T.G.: The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452 (1981)

    MATH  Article  MathSciNet  Google Scholar 

  5. 5

    Feng S., Wang F.Y.: A class of infinite-dimensional diffusion processes with connection to population genetics. J. Appl. Prob. 44, 938–949 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6

    Ferguson T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)

    MATH  Article  MathSciNet  Google Scholar 

  7. 7

    Fukushima M., Oshima Y., Takeda M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin/New York (1994)

    MATH  Google Scholar 

  8. 8

    Griffiths R.C.: A transition density expansion for a multi-allele diffusion model. Adv. Appl. Prob. 11, 310–325 (1979)

    MATH  Article  Google Scholar 

  9. 9

    Handa, K.: The two-parameter Poisson-Dirichlet point process. Bernoulli (to appear) (2009)

  10. 10

    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)

    MATH  Article  Google Scholar 

  11. 11

    Kingman J.C.F.: Random discrete distributions. J. Roy. Statist. Soc. B. 37, 1–22 (1975)

    MATH  MathSciNet  Google Scholar 

  12. 12

    Lamperti J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc. 88, 380–387 (1958)

    MATH  MathSciNet  Google Scholar 

  13. 13

    Ma Z.M., Röckner M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer-Verlag, Berlin (1992)

    MATH  Google Scholar 

  14. 14

    Mosco U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, 368–421 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  15. 15

    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)

    MATH  Article  Google Scholar 

  16. 16

    Petrov, L.: A two-parameter family of infinite-dimensional diffusions in the Kingman simplex. http://arxiv.org/abs/0708.1930 (2007)

  17. 17

    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)

  18. 18

    Pitman, J.: Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer (2006)

  19. 19

    Pitman J., Yor M.: The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  20. 20

    Schmuland B.: A result on the infinitely many neutral alleles diffusion model. J. Appl. Prob. 28, 253–267 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  21. 21

    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)

  22. 22

    Schmuland, B.: Lecture Notes on Dirichlet Forms. http://www.stat.ualberta.ca/people/schmu/preprints/yonsei.pdf (1995)

  23. 23

    Watterson G.A.: The stationary distribution of the infinitely many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651 (1976)

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Shui Feng.

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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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Mathematics Subject Classification (2000)

  • Primary: 60F10
  • Secondary: 92D10