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The Poisson-Kingman distributions, $\mathrm{PK}(\rho)$, on the infinitesimplex, can be constructed from a Poisson point process having intensitydensity $\rho$ or by taking the ranked jumps up till a specified time of asubordinator with L\'evy density $\rho$, as proportions of the subordinator. Asa natural extension, we replace the Poisson point process with a negativebinomial point process having parameter $r>0$ and L\'evy density $\rho$,thereby defining a new class $\mathrm{PK}^{(r)}(\rho)$ of distributions on theinfinite simplex. The new class contains the two-parameter generalisation$\mathrm{PD}(\alpha, \theta)$ of Pitman and Yor (1997) when $\theta>0$. It alsocontains a class of distributions derived from the trimmed stable subordinator.We derive properties of the new distributions, with particular reference to thetwo most well-known $\mathrm{PK}$ distributions: the Poisson-Dirichletdistribution $\mathrm{PK}(\rho_\theta)$ generated by a Gamma process withL\'evy density $\rho_\theta(x) = \theta e^{-x}/x$, $x>0$, $\theta > 0$, and therandom discrete distribution, $\mathrm{PD}(\alpha,0)$, derived from an$\alpha$-stable subordinator.

Negative Binomial Construction of Random Discrete Distributions on the Infinite Simplex