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When $S=(S_t)_{t\ge 0}$ is an $\alpha$-stable subordinator, the sequence ofordered jumps of $S$, up till time $1$, omitting the $r$ largest of them, andtaken as proportions of their sum $^{(r)}S_t$, defines a 2-parameterdistribution on the infinite dimensional simplex, $\nabla_{\infty}$, which wecall the $\mathrm{PD}_\alpha^{(r)}$ distribution. When $r=0$ it reduces to the$\mathrm{PD}_\alpha$ distribution introduced by Kingman in 1975. We observe aserendipitous connection between $\mathrm{PD}_\alpha^{(r)}$ and the negativebinomial point process of Gregoire (1984), which we exploit to analyse indetail a size-biased version of $\mathrm{PD}_\alpha^{(r)}$. As a consequence wederive a stick-breaking representation for the process and a useful form forits distribution. This program produces a large new class of distributionsavailable for a variety of modelling purposes.

Generalised Poisson-Dirichlet Distributions and the Negative Binomial Point Process