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April 1997 The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
Jim Pitman, Marc Yor
Ann. Probab. 25(2): 855-900 (April 1997). DOI: 10.1214/aop/1024404422

Abstract

The two-parameter Poisson-Dirichlet distribution, denoted $\mathsf{PD}(\alpha, \theta)$ is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter $\theta$, introduced by Kingman, is $\mathsf{PD}(0, \theta)$. Known properties of $\mathsf{PD}(0, \theta)$, including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of $\mathsf{PD}(\alpha, \theta)$ is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For $0 < \alpha < 1, \mathsf{PD}(\alpha, 0)$ is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index $\alpha$. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is $\mathsf{PD}(1/2, 0)$, and the corresponding distribution for a Brownian bredge is $\mathsf{PD}(1/2, 1/2)$. The $\mathsf{PD}(\alpha, 0)$ and $\mathsf{PD}(\alpha, \alpha)$ distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index $\alpha$.

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Jim Pitman. Marc Yor. "The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator." Ann. Probab. 25 (2) 855 - 900, April 1997. https://doi.org/10.1214/aop/1024404422

Information

Published: April 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0880.60076
MathSciNet: MR1434129
Digital Object Identifier: 10.1214/aop/1024404422

Subjects:
Primary: 60E99, 60G57, 60J30

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 2 • April 1997
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