{
  "id": "0711.3686",
  "version": "v4",
  "published": "2007-11-23T09:37:11.000Z",
  "updated": "2010-11-17T14:46:46.000Z",
  "title": "Biased random walks on a Galton-Watson tree with leaves",
  "authors": [
    "Gérard Ben Arous",
    "Alexander Fribergh",
    "Nina Gantert",
    "Alan Hammond"
  ],
  "comment": "49 pages, 2 figures. To appear in Ann. Probab",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\\gamma= \\gamma(\\beta) \\in (0,1)$, depending on the bias $\\beta$, such that $X_n$ is of order $n^{\\gamma}$. Denoting $\\Delta_n$ the hitting time of level $n$, we prove that $\\Delta_n/n^{1/\\gamma}$ is tight. Moreover we show that $\\Delta_n/n^{1/\\gamma}$ does not converge in law (at least for large values of $\\beta$). We prove that along the sequences $n_{\\lambda}(k)=\\lfloor \\lambda \\beta^{\\gamma k}\\rfloor$, $\\Delta_n/n^{1/\\gamma}$ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2010-11-17T14:46:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "biased random walk",
      "galton-watson tree",
      "sub-ballistic regime",
      "explicit constant",
      "large values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.3686B"
    }
  }
}