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 L\'evy 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 L\'evy 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 L\'evy 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.
Categories: math.PR