Limits of P\'olya urns with innovations

TL;DR

This paper studies a generalized Pólya urn model with innovations, analyzing its long-term behavior and fluctuations, especially focusing on the impact of the ratio of returned balls on convergence.

Contribution

It introduces a Pólya urn scheme with innovations on arbitrary measurable color spaces and characterizes its convergence and fluctuations based on the ratio of returned to total balls.

Findings

Empirical distribution converges to the normalized intensity measure of the innovation process.

The ratio of expected copies to total returned balls influences the fluctuation behavior.

The model generalizes classical Pólya urns to arbitrary color spaces with innovations.

Abstract

We consider a version of the classical P\'olya 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.