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

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.