RESEARCH ARTICLE

Gamma-Dirichlet algebra and applications

  • Shui FENG , 1 ,
  • Fang XU 2
Expand
  • 1. Department of Mathematics and Statistics, McMaster University, Hamilton, Ont L8S 4K1, Canada
  • 2. Canadian Imperial Bank of Commerce, 21 Melinda St, Toronto, Ont M5L 1B9, Canada

Received date: 09 Jan 2014

Accepted date: 08 Jun 2014

Published date: 20 Aug 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

The Gamma-Dirichlet algebra corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief survey of several existing results concerning this structure. New results are then obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. We finish the paper with the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration by exploring the Gamma-Dirichlet algebra embedded in these processes. This last result is motivated by an open problem proposed by S. N. Ethier and R. C. Griffiths.

Cite this article

Shui FENG , Fang XU . Gamma-Dirichlet algebra and applications[J]. Frontiers of Mathematics in China, 2014 , 9(4) : 797 -812 . DOI: 10.1007/s11464-014-0408-0

1
Dawson D A, Feng S. Asymptotic behavior of Poisson-Dirichlet distribution for large mutation rate. Ann Appl Probab, 2006, 16(2): 562-582

DOI

2
Dembo A, Zeitouni O. Large Deviations Techniques and Applications. 2nd ed. Applications of Mathematics, Vol 38. New York: Springer-Verlag, 1998

3
Ethier S N, Griffiths R C. The transition function of a Fleming-Viot process. Ann Probab, 1993, 21(3): 1571-1590

DOI

4
Ethier S N, Griffiths R C. The transition function of a measure-valued branching diffusion with immigration. In: Cambanis S, Ghosh J, Karandikar R L, Sen P K, eds. Stochastic Processes. A Festschrift in Honour of Gopinath Kallianpur. New York: Springer, 1993, 71-79

5
Ethier S N, Kurtz T G. Convergence to Fleming-Viot processes in the weak atomic topology. Stochastic Process Appl, 1994, 54: 1-27

DOI

6
Feng S. Poisson-Dirichlet distribution with small mutation rate. Stochastic Process Appl, 2009, 119: 2082-2094

DOI

7
Feng S. The Poisson-Dirichlet Distribution and Related Topics. Probability and Its Applications. New York: Springer, 2010

DOI

8
Griffiths R C. On the distribution of allele frequencies in a diffusion model. Theor Pop Biol, 1979, 15: 140-158

DOI

9
Handa K. Quasi-invariant measures and their characterization by conditional probabilities. Bull Sci Math, 2001, 125(6-7): 583-604

DOI

10
Kingman J C F. Random discrete distributions. J R Stat Soc B, 1975, 37: 1-22

11
Li Z H. Measure-Valued Branching Markov Processes. Probability and Its Applications. New York: Springer, 2011

DOI

12
Lukacs E. A characterization of the gamma distribution. Ann Math Statist, 1955, 26: 319-324

DOI

13
Perman M, Pitman J, Yor M. Size-biased sampling of Poisson point processes and excursions. Probab Theory Related Fields, 1992, 92: 21-39

DOI

14
Pitman J, Yor M. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann Probab, 1997, 25(2): 855-900

DOI

15
Shiga T. A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J Math Kyoto Univ, 1990, 30: 245-279

16
Tavaré S. Line-of-descent and genealogical processes, and their applications in population genetics models. Theor Pop Biol, 1984, 26: 119-164

DOI

17
Tsilevich N V, Vershik A. Quasi-invariance of the gamma process and the multiplicative properties of the Poisson-Dirichlet measures. C R Acad Sci Paris, Sér I, 1999, 329: 163-168

18
Tsilevich N V, Vershik A, Yor M. An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process. J Funct Anal, 2001, 185(1): 274-296

DOI

Outlines

/