Fluctuations of balanced urns with infinitely many colours

In this paper, we prove convergence and fluctuation results for measure-valued P\'olya processes (MVPPs, also known as P\'olya urns with infinitely-many colours). Our convergence results hold almost surely and in $L^2$, under assumptions that are different from that of other convergence results in the literature. 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 P\'olya urns. As in the finitely-many-colour 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.