{
  "id": "0912.1392",
  "version": "v1",
  "published": "2009-12-08T03:07:00.000Z",
  "updated": "2009-12-08T03:07:00.000Z",
  "title": "Consistent Minimal Displacement of Branching Random Walks",
  "authors": [
    "Ming Fang",
    "Ofer Zeitouni"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\mathbb{T}$ denote a rooted $b$-ary tree and let $\\{S_v\\}_{v\\in \\mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $\\Lambda(\\cdot)$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by $\\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a constant, which for convenience may be taken to be 0. With $\\bar S_v=\\max\\{S_w:{\\rm $w$ is on the geodesic connecting the root to $v$}\\}$, define $L_n=\\min_{v\\in \\mathbb{D}_n} \\bar S_v$. We prove that $L_n/n^{1/3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-08T03:07:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60J80"
    ],
    "keywords": [
      "branching random walk",
      "consistent minimal displacement",
      "logarithmic moment generating function",
      "explicit constant",
      "ary tree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.1392F"
    }
  }
}