Negative Binomial Construction of Random Discrete Distributions on the Infinite Simplex

Abstract

The Poisson-Kingman distributions, PK(ρ), on the infinite simplex, can be constructed from a Poisson point process having intensity density ρ or by taking the ranked jumps up till a specified time of a subordinator with Lévy density ρ, as proportions of the subordinator. As a natural extension, we replace the Poisson point process with a negative binomial point process having parameter r > 0 and Lévy density ρ, thereby defining a new class PK(r)(ρ) of distributions on the infinite simplex. The new class contains the two-parameter generalisation PD(α, θ) of [13] when θ > 0. It also contains a class of distributions derived from the trimmed stable subordinator. We derive properties of the new distributions, with particular reference to the two most well-known PK distributions: the Poisson-Dirichlet distribution PK(ρθ) generated by a Gamma process with Lévy density ρθ(x) = θe−x/x, x > 0, θ > 0, and the random discrete distribution, PD(α, 0), derived from an α-stable subordinator.