Negative Binomial Construction of Random Discrete Distributions on the Infinite Simplex

Abstract

The Poisson-Kingman distributions, $\mathrm{PK}(\rho )$, on the infinite simplex, can be constructed from a Poisson point process having intensity density $\rho$ or by taking the ranked jumps up till a specified time of a subordinator with Lévy density $\rho$, 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 $\rho$, thereby defining a new class ${\mathrm{PK}}^{(r)}(\rho )$ of distributions on the infinite simplex. The new class contains the two-parameter generalisation $\mathrm{PD}(\alpha ,\theta )$ of Pitman and Yor (1997) when $\theta >0$. 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 $\mathrm{PK}$ distributions: the Poisson-Dirichlet distribution $\mathrm{PK}({\rho }_{\theta })$ generated by a Gamma process with Lévy density ${\rho }_{\theta }(x)=\theta e^{-x}/x$, $x>0$, $\theta >0$, and the random discrete distribution, $\mathrm{PD}(\alpha ,0)$, derived from an $\alpha$-stable subordinator.

This paper includes the main result of arXiv:1611.09980 with a different proof and further generalisation