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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Feng, S., Xu, F. Gamma-Dirichlet algebra and applications. Front. Math. China 9, 797–812 (2014). https://doi.org/10.1007/s11464-014-0408-0
- Dirichlet process
- gamma process
- random time-change