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 $\xi$ on $S$. When the number of steps goes to infinity, the empirical distribution of the colors in the urn converges to the normalized intensity measure of $\xi$, and we analyze the fluctuations. The ratio $\rho =E(C)/E(R)$ of the average number of copies to the average total number of balls returned plays a key role.

We consider much more general replacement scheme compared to the first version. The guiding lines are similar; nonetheless several steps require new arguments