Limits of Pólya urns with innovations

Abstract

We consider a version of the classical Pólya urn scheme which incorporates innovations. The space S of colors is an arbitrary measurable set. After each sampling of a ball in the urn, one returns C balls of the same color and additional balls of different colors given by some finite point process ξ on S, where the distribution $P_s$ of the pair $(C,\mathrm{\xi })$ depends on the sampled color s. We suppose that the average number of copies $E_s(C)$ is the same for all $s\in S$, and that the intensity measures of innovations have the form $E_s(\mathrm{\xi })=a(s)\mathrm{\mu }$ for some finite measure μ and a modulation function a on S that is bounded away from 0 and ∞. We then show that the empirical distribution of the colors in the urn converges to the normalized intensity $\bar{\mathrm{\mu }}$. In turn, different regimes for the fluctuations are observed, depending on whether $E(C)$ is larger or smaller than $\mathrm{\mu }(a)$.