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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 ,
A. D. BARBOUR  and
SIMON TAVARÉ
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)
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Abstract

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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.

Information

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 8 , Issue 5 , September 1999 , pp. 407 - 416
Copyright
1999 Cambridge University Press