Measure-valued Pólya urn processes

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,… ,d\ for d∈ 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, Rc,j balls of colour j are also added (for all 1≤ j≤ 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 P. We see the urn composition at any time step n as a measure Mₙ – possibly non atomic – on P. In this generalisation, we choose a random colour c according to the probability distribution proportional to Mₙ, and add a measure Rc in the urn, where the quantity Rc(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 ( Rc,c∈ P) for the sequence of measures ( Mₙ, n≥ 0) to converge in distribution, possibly after rescaling. For certain models, related to branching random walks, ( Mₙ, n≥ 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.