Poisson representation of a Ewens fragmentation process

Abstract

A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of $[n]=\{1,...,n\}$ at time $\theta \ge 0$ is governed by the Ewens sampling formula with parameter $\theta$. These partition-valued processes are exchangeable and consistent, as $n$ varies. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity $\theta x^{-1} \diff x$ on ${\mathbb R}_+$, arranged to be intensifying as $\theta$ increases.