The Dirichlet process has been extensively studied over the last thirty years, along with various generalisations, and remains a fundamental 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 together, using a new limit type representation of the Dirichlet process. This in particular allows simpler derivation of some of the previous results in the literature. Some new results are also reached.