The Poisson–Dirichlet Distribution and the Scale-Invariant Poisson Process
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)
1 Supported in part by NSF grant DMS 96-26412.
2 Supported in part by Schweizerischer NF Projekt Nr 20-43453.95.