{
  "id": "math/0508616",
  "version": "v1",
  "published": "2005-08-30T14:50:24.000Z",
  "updated": "2005-08-30T14:50:24.000Z",
  "title": "Fragmentation processes with an initial mass converging to infinity",
  "authors": [
    "Bénédicte Haas"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let $F\\_{1}^{(m)}(t),F\\_{2}^{(m)}(t),...$ denote the decreasing rearrangement of the masses present at time $t$ in a such process, starting from an initial mass $m$. Let then $m\\to \\infty $. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence $(F\\_{2}^{(m)},F\\_{3}^{(m)},...)$ converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of $m-F\\_{1}^{(m)}$ to a stable subordinator. A continuum random tree counterpart of this result is also given: the continuum random tree describing the genealogy of a self-similar fragmentation satisfying the required assumption and starting from a mass converging to $\\infty $ will converge to a tree with a spine coding a fragmentation with immigration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-08-30T14:50:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J25",
      "60F05"
    ],
    "keywords": [
      "fragmentation processes",
      "initial mass converging",
      "continuum random tree counterpart",
      "regular variation type",
      "immigration process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8616H"
    }
  }
}