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The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. (1995)
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@MISC{Pitman95thetwo-parameter,
author = {Jim Pitman and Marc Yor},
title = {The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator.},
year = {1995}
}
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Abstract
The two-parameter Poisson-Dirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to Vershik-Shmidt-Ignatov, are generalized to the two-parameter case. The size-biased random permutation of pd(ff; `) 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 ! ff ! 1, pd(ff; 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 ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Keyphrases
two-parameter poisson-dirichlet distribution stable subordinator ranked length usual poisson-dirichlet distribution markov chain asymptotic distribution specie diversity simple residual allocation model recurrence time distribution bessel process stable law index ff single parameter size-biased random permutation positive sequence case trace brownian motion two-parameter case markov chain description
