Generalised Poisson-Dirichlet Distributions and the Negative Binomial Point Process

Abstract

When $S=(S_t{)}_{t\ge 0}$ is an $\alpha$-stable subordinator, the sequence of ordered jumps of $S$, up till time $1$, omitting the $r$ largest of them, and taken as proportions of their sum ${}^{(r)}{S}_t$, defines a 2-parameter distribution on the infinite dimensional simplex, ${\nabla }_{\infty }$, which we call the ${\mathrm{PD}}_{\alpha }^{(r)}$ distribution. When $r=0$ it reduces to the ${\mathrm{PD}}_{\alpha }$ distribution introduced by Kingman in 1975. We observe a serendipitous connection between ${\mathrm{PD}}_{\alpha }^{(r)}$ and the negative binomial point process of Gregoire (1984), which we exploit to analyse in detail a size-biased version of ${\mathrm{PD}}_{\alpha }^{(r)}$. As a consequence we derive a stick-breaking representation for the process and a useful form for its distribution. This program produces a large new class of distributions available for a variety of modelling purposes.

17 pages

References in corpus (2)

• The limit distribution of ratios of jumps and sums of jumps of subordinators
• On financial applications of the two-parameter Poisson-Dirichlet distribution

Cited by in corpus (1)

• Models for Genetic Diversity Generated by Negative Binomial Point Processes