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Gamma-Dirichlet algebra and applications |
Shui FENG1,*( ),Fang XU2 |
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 |
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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.
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| Keywords
Coalescent
Dirichlet process
gamma process
quasi-invariant
random time-change
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Corresponding Author(s):
Shui FENG
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Issue Date: 26 August 2014
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