Fluctuations of balanced urns with infinitely many colours

In this paper, we prove convergence and fluctuation results for measure-valued Polya processes (MVPPs, also known as Polya urns with infinitely-many colours). Our convergence results hold almost surely and in L2. Our fluctuation results are the first second-order results in the literature on MVPPs; they generalise classical fluctuation results from the literature on finitely-many-colour Polya urns. As in the finitely-manycolour case, the order and shape of the fluctuations depend on whether the "spectral gap is small or large". To prove these results, we show that MVPPs are stochastic approximations taking values in the set of measures on a measurable space E (the colour space). We then use martingale methods and standard operator theory to prove convergence and fluctuation results for these stochastic approximations.