Article,
Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex
Stochastic Process. Appl., 31 (1): 89--103 (1989)
Abstract
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.
