{
  "id": "1610.09057",
  "version": "v1",
  "published": "2016-10-28T02:13:01.000Z",
  "updated": "2016-10-28T02:13:01.000Z",
  "title": "Measure-valued Pólya processes",
  "authors": [
    "Cécile Mailler",
    "Jean-François Marckert"
  ],
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-28T02:13:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "measure-valued pólya processes",
      "time step",
      "measure-valued polya urn processes",
      "probability distribution proportional",
      "generalisation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}