Unit-rate Poisson representations of completely random measures

Constructive definitions of discrete random measures, which specify a sampling procedure for the weights and atom locations of the measure, have proven to be of great value in statistics and related fields. We consider the case of completely random measures and obtain a constructive representation for completely random measures on Euclidean space. For random measures on the real line satisfying a specific σ-finiteness property, the representation is equivalent to the Ferguson-Klass representation of pure-jump Lévy processes. As examples, we provide ”stick-breaking” representations of the gamma process, the stable process and the beta process.