{
  "id": "1004.3061",
  "version": "v1",
  "published": "2010-04-18T21:44:57.000Z",
  "updated": "2010-04-18T21:44:57.000Z",
  "title": "Growth of Galton-Watson trees: immigration and lifetimes",
  "authors": [
    "Xiao'ou Cao",
    "Matthias Winkel"
  ],
  "comment": "31 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study certain consistent families $(F_\\lambda)_{\\lambda\\ge 0}$ of Galton-Watson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in $F_\\lambda$. Specifically, consistency here refers to the property that for each $\\mu\\le\\lambda$, the forest $F_\\mu$ has the same distribution as the subforest of $F_\\lambda$ spanned by the black leaves in a Bernoulli leaf colouring, where each leaf of $F_\\lambda$ is coloured in black independently with probability $\\mu/\\lambda$. The case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuous-state branching processes. We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, related to Sagitov's (non-Markovian) generalisation of continuous-state branching renewal processes, and similar processes with immigration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-18T21:44:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80"
    ],
    "keywords": [
      "galton-watson trees",
      "immigration",
      "continuous-state branching renewal processes",
      "arbitrary lifetime distributions",
      "markovian continuous-state branching processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.3061C"
    }
  }
}