Abstract
A P\'olya 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 measurevalued 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 ${\mathcal M}_n$  possibly non atomic  on $\mathcal P$. In this generalisation, we choose a random colour $c$ according to the probability distribution proportional to ${\mathcal M}_n$, and add a measure ${\mathcal R}_c$ in the urn, where the quantity ${\mathcal R}_c(B)$ of a Borelian $B$ models the added weight of "balls" with colour in $B$. We study the asymptotic behaviour of these measurevalued P\'olya urn processes, and give some conditions on the replacements measures $({\mathcal R}_c, c\in \mathcal P)$ for the sequence of measures $({\mathcal M}_n, n\geq 0)$ to converge in distribution after a possible rescaling. For certain models, related to branching random walks, $({\mathcal M}_n, n\geq 0)$ is shown to converge almost surely under some moment hypothesis.
Original language  English 

Article number  26 
Number of pages  33 
Journal  Electronic Journal of Probability 
Volume  22 
DOIs  
Publication status  Published  21 Mar 2017 
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Cecile Mailler
 Department of Mathematical Sciences  Reader
 Probability Laboratory at Bath
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching, Researcher