Measure-valued Pólya urn processes

Abstract

A Pólya urn process is a Markov chain that models the evolution of an urn containing some coloured balls, the set of possible colours being $\{1,\ldots ,d\}$ for $d\in \mathbb{N} $. At each time step, a random ball is chosen uniformly in the urn. It is replaced in the urn and, if its colour is $c$, $R_{c,j}$ balls of colour $j$ are also added (for all $1\leq j\leq d$).

We introduce a model of measure-valued processes that generalises this construction. This generalisation includes the case when the space of colours is a (possibly infinite) Polish space $\mathcal P$. We see the urn composition at any time step $n$ as a measure ${\cal M}_n$ – possibly non atomic – on $\mathcal P$. In this generalisation, we choose a random colour $c$ according to the probability distribution proportional to ${\cal M}_n$, and add a measure ${\cal R}_c$ in the urn, where the quantity ${\cal R}_c(B)$ of a Borel set $B$ models the added weight of “balls” with colour in $B$.

We study the asymptotic behaviour of these measure-valued Pólya urn processes, and give some conditions on the replacements measures $({\cal R}_c,c\in \mathcal P)$ for the sequence of measures $({\cal M}_n, n\geq 0)$ to converge in distribution, possibly after rescaling. For certain models, related to branching random walks, $({\cal M}_n, n\geq 0)$ is shown to converge almost surely under some moment hypothesis; a particular case of this last result gives the almost sure convergence of the (renormalised) profile of the random recursive tree to a standard Gaussian.