On the distribution of random Dirichlet jumps

The Dirichlet process has been extensively studied over the last thirty years, along with various generalisations, and remains a fundemental tool for nonparametric Bayesian statistics. The probabilistic structure of its jumps has not drawn so much attention in those contexts, however, but has been examined in somewhat unrelated literature, ranging from probabilistic number theory, population genetics, mathematical ecology, and size-biased sampling theory. This paper connects some of these theories and results, using a limit type characterisation of the Dirichlet process and an invariance property of a particular representation of its jumps. These in particular allow simpler derivations of some of the previous results in the literature. Some new results are also reached.