Multiple drawing multi-colour urns by stochastic approximation

Abstract

A classical P\'olya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$). We are interested in multi-drawing P\'olya urns, where the replacement rule depends on the random drawing of a set of $m$ balls from the urn (with or without replacement). This generalisation has already been studied in the literature, in particular by Kuba & Mahmoud (ArXiv:1503.09069 and 1509.09053), where second order asymptotic results are proved for $2$-colour urns under the balanced and the affinity assumptions. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to remove the affinity hypothesis of Kuba & Mahmoud and generalise the result to more-than-two-colour urns. We also give some partial results in the two-colour non-balanced case.