Generalized Polya urns via stochastic approximation

TL;DR

This paper develops stochastic approximation methods to analyze generalized Polya urn models, determining their limiting behavior and extending classical urn results to more complex drawing and replacement schemes.

Contribution

It introduces a unified stochastic approximation framework for broad generalizations of Polya urns, including models with multiple draws and complex replacement rules.

Findings

Derived limiting fractions for generalized urn models

Extended classical Polya urn results to new complex schemes

Validated methods through specific model examples

Abstract

We collect, survey and develop methods of (one-dimensional) stochastic approximation in a framework that seems suitable to handle fairly broad generalizations of Polya urns. To show the applicability of the results we determine the limiting fraction of balls in an urn with balls of two colors. We consider two models generalizing the Polya urn, in the first one ball is drawn and replaced with balls of (possibly) both colors according to which color was drawn. In the second, two balls are drawn simultaneously and replaced along with balls of (possibly) both colors according to what combination of colors were drawn.