Combinatorics, Probability and Computing



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


RICHARD ARRATIA a1 1 , A. D. BARBOUR a2 2 and SIMON TAVARÉ a1 1
a1 Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA (e-mail: rarratia@math.usc.edu stavare@gnome.usc.edu)
a2 Abteilung für Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland (e-mail: adb@amath.unizh.ch)

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[less-than-or-eq, slant]1. Restricting both processes to (0, β] for 0<β[less-than-or-eq, slant]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.

(Received June 27 1997)
(Revised March 16 1998)



Footnotes

1 Supported in part by NSF grant DMS 96-26412.

2 Supported in part by Schweizerischer NF Projekt Nr 20-43453.95.