Multi-drawing Pólya urns via labelled random DAGs

Abstract

A Pólya urn of replacement matrix $R=(R_{i,j})_{1\leq i,j\leq d}$ is a Markov process that encodes the following experiment: an urn contains balls of $d$ different colours and at every time-step, a ball is drawn uniformly at random in the urn, and if its colour is $i$, then it is replaced in the urn with an additional $R_{i,j}$ balls of colour $j$, for all $1\leq i, j\leq d$. We study a natural extension of this model in which, instead of drawing one ball at each time-step, we draw a set of $m\geq 2$ balls: in this case, the replacement matrix becomes a replacement tensor. Because of the multi-draws, this process can no longer be seen as a branching process, which makes its analysis much more intricate than in the classical Pólya urn case. Partial results proved by stochastic approximation techniques exist in the literature. In this article, we introduce a new approach based on seeing the process as a stochastic process indexed by a random directed-acyclic graph (DAG) and use this approach, together with the theory of stochastic tensors, to prove a convergence theorem for these multi-drawing Pólya urns, with assumptions that are straightforward to check in practice.