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The Poisson–Dirichlet Distribution and the Scale-Invariant Poisson Process

Published online by Cambridge University Press:  01 September 1999

RICHARD ARRATIA
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA (e-mail: rarratia@math.usc.edu stavare@gnome.usc.edu)
A. D. BARBOUR
Affiliation:
Abteilung für Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland (e-mail: adb@amath.unizh.ch)
SIMON TAVARÉ
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA (e-mail: rarratia@math.usc.edu stavare@gnome.usc.edu)

Abstract

We show that the Poisson–Dirichlet distribution is the distribution of points in a scale-invariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that T[les ]1. Restricting both processes to (0, β] for 0<β[les ]1, we give an explicit formula for the total variation distance between their distributions. Connections between various representations of the Poisson–Dirichlet process are discussed.

Type
Research Article
Copyright
1999 Cambridge University Press

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