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 6 1. Restricting both processes to (0, β] for 0 < β 6 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. 1. The Poisson–Dirichlet process This paper gives a new characterization of the Poisson–Dirichlet distribution, showing its relation with the scale-invariant Poisson process. The Poisson–Dirichlet process (V 1 , V 2 ,. . .) with parameter θ > 0 (Kingman [15, 16], Watterson [25]) plays a fundamental role in combinatorics and number theory: see the exposition in [3]. The coordinates satisfy V 1 > V 2 > · · · > 0 and V 1 + V 2 + · · · = 1 almost surely. The distribution of this process is most directly characterized by the density functions of its finite-dimensional distributions. The joint density of (V 1 , V 2 , · · · , V k) is supported by points