A Recursive Method for Functionals of Poisson Processes

Functionals of Poisson processes arise in many statistical problems. They appear in problems involving heavy-tailed distributions in the study of limiting processes, while in Bayesian nonparametric statistics they are used as constructive representations for nonparametric priors. We describe a simple recursive method that is useful for characterizing Poisson process functionals that requires only the use of conditional probability. Applications of this technique to convex hulls, extremes, stable measures, infinitely divisible random variables and Bayesian nonparametric priors are discussed.