{
  "id": "1102.0217",
  "version": "v4",
  "published": "2011-02-01T17:02:48.000Z",
  "updated": "2014-04-04T12:50:22.000Z",
  "title": "The Seneta--Heyde scaling for the branching random walk",
  "authors": [
    "Elie Aidekon",
    "Zhan Shi"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/12-AOP809 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2014, Vol. 42, No. 3, 959-993",
  "doi": "10.1214/12-AOP809",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609--631] in a one-dimensional super-critical branching random walk, and study the additive martingale $(W_n)$. We prove that, upon the system's survival, $n^{1/2}W_n$ converges in probability, but not almost surely, to a positive limit. The limit is identified as a constant multiple of the almost sure limit, discovered by Biggins and Kyprianou [Adv. in Appl. Probab. 36 (2004) 544--581], of the derivative martingale.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-04-04T12:50:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "seneta-heyde scaling",
      "one-dimensional super-critical branching random walk",
      "sure limit",
      "boundary case",
      "systems survival"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.0217A"
    }
  }
}