On the stick-breaking representation of sigma-stable Poisson-Kingman models

In this paper we investigate the stick-breaking representation for the class of σ-stable Poisson-Kingman models, also known as Gibbs-type random probability measures. This class includes as special cases most of the discrete priors commonly used in Bayesian nonparametrics, such as the two parameter Poisson-Dirichlet process and the normalized generalized Gamma process. Under the assumption σ=u/v, for any coprime integers 1≤u<v such that u/v≤1/2, we show that a σ-stable Poisson-Kingman model admits an explicit stick-breaking representation in terms of random variables which are obtained by suitably transforming Gamma random variables and products of independent Beta and Gamma random variables.