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