Fluctuations of balanced urns with infinitely many colours

Abstract

In this paper, we prove convergence and fluctuation results for measurevalued Pólya processes (MVPPs, also known as Pólya 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 Pólya 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.

Limits of Pólya urns with innovations

“…Fluctuations for measure-valued Pólya processes have been considered very recently in [14], where results bearing the same flavor as our Theorem 5.1 are established. However [14] imposes a strong balance condition that would imply in our case that R ≡ 1. Thus, although general urn schemes with random replacement kernels encompass the setting of the present work and Theorem 1.1 is a close relative to results in [14], the present contribution cannot be reduced to [14] either.…”

“…In particular the observation (1.4) at the origin of this work can be seen as a special case of [19,Theorem 1] or [13,Section 6], or also [3]. Fluctuations for measure-valued Pólya processes have been considered very recently in [14], where results bearing the same flavor as our Theorem 5.1 are established. However [14] imposes a strong balance condition that would imply in our case that R ≡ 1.…”

“…However [14] imposes a strong balance condition that would imply in our case that R ≡ 1. Thus, although general urn schemes with random replacement kernels encompass the setting of the present work and Theorem 1.1 is a close relative to results in [14], the present contribution cannot be reduced to [14] either.…”

“…Keep in mind that the reinforcement variable C also impacts λ 1 and λ 2 , and in particular that increasing C while keeping the innovation ξ unchanged also increases the ratio ρ. It has been briefly mentioned at the beginning of this introduction that urns schemes with infinite colors, or measure-valued Pólya processes, have been considered in several recent works [3,4,5,13,14,18,19], and we shall now recast our contribution from this perspective. In [5] and [18], the authors consider urns processes as Markov chains (U n ) n≥0 of random measures (not necessarily point processes) on a measurable space S, such that conditionally given U 0 , .…”