### Abstract

Language | English |
---|---|

Article number | 26 |

Number of pages | 33 |

Journal | Electronic Journal of Probability |

Volume | 22 |

DOIs | |

Status | Published - 21 Mar 2017 |

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### Cite this

*Electronic Journal of Probability*,

*22*, [26]. https://doi.org/10.1214/17-EJP47

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

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 22, 26. https://doi.org/10.1214/17-EJP47

}

TY - JOUR

T1 - Measure-valued Pólya processes

AU - Mailler, Cécile

AU - Marckert, Jean-François

PY - 2017/3/21

Y1 - 2017/3/21

N2 - 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.

AB - 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.

UR - http://dx.doi.org/10.1214/17-EJP47

UR - http://dx.doi.org/10.1214/17-EJP47

U2 - 10.1214/17-EJP47

DO - 10.1214/17-EJP47

M3 - Article

VL - 22

JO - Electronic Journal of Probability

T2 - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 26

ER -