Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex

Ranked and size-biased permutations are particular functions on the set of probability measures on the simplex. They represent two recently studied schemes for relabelling groups in certain stochastic models, and are of particular interest in describing the limiting behaviour of such models. We prove that the ranked permutations of a sequence of measures converge if and only if the size-biased permutations converge, and give conditions under which weak convergence of measures guarantees weak convergence of both permutations. Applications include a proof of the fact that the GEM distribution is the size biased permutation of the Poisson-Dirichlet and a new proof of the fact that when labelled in a particular way, normalized cycle lengths in a random permutation converge to the GEM distribution. These techniques also allow some problems concerned with the random splitting of an interval to be related to known results in other fields.