{
  "id": "math/0604439",
  "version": "v1",
  "published": "2006-04-20T06:46:48.000Z",
  "updated": "2006-04-20T06:46:48.000Z",
  "title": "Regular variation in the branching random walk",
  "authors": [
    "Aleksander Iksanov",
    "Sergey Polotskiy"
  ],
  "comment": "submitted",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{\\mm_n, n=0,1,...\\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,...$ let $W_n$ be the moment generating function of $\\mm_n$ normalized by its mean. Denote by $AW_n$ any of the following random variables: maximal function, square function, $L_1$ and a.s. limit $W$, $\\su |W-W_n|$, $\\su |W_{n+1}-W_n|$. Under mild moment restrictions and the assumption that $\\rP\\{W_1>x\\}$ regularly varies at $\\infty$ it is proved that $\\rP\\{AW_n>x\\}$ regularly varies at $\\infty$ with the same exponent. All the proofs given are non-analytic in the sense that these do not use Laplace-Stieltjes transforms. The result on the tail behaviour of $W$ is established in two distinct ways.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-04-20T06:46:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G42",
      "60J80",
      "60E99"
    ],
    "keywords": [
      "regular variation",
      "branching random walk starting",
      "mild moment restrictions",
      "regularly varies",
      "tail behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......4439I"
    }
  }
}