# Measure-valued Pólya processes

Cécile Mailler, Jean-François Marckert

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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 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 ${\mathcal M}_n$ -- 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 measure-valued 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 26 33 Electronic Journal of Probability 22 https://doi.org/10.1214/17-EJP47 Published - 21 Mar 2017

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Ball
Measure-valued Process
Converge
Branching Random Walk
Polish Space
Markov Chain Model
Rescaling
Replacement
Color
Probability Distribution
Choose
Asymptotic Behavior
Directly proportional
Model
Moment
Generalise
Generalization

### Cite this

Measure-valued Pólya processes. / Mailler, Cécile; Marckert, Jean-François.

In: Electronic Journal of Probability, Vol. 22, 26, 21.03.2017.

Research output: Contribution to journalArticle

Mailler, Cécile ; Marckert, Jean-François. / Measure-valued Pólya processes. In: Electronic Journal of Probability. 2017 ; Vol. 22.
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