{
  "id": "1106.4281",
  "version": "v1",
  "published": "2011-06-21T18:02:45.000Z",
  "updated": "2011-06-21T18:02:45.000Z",
  "title": "Convergence to type I distribution of the extremes of sequences defined by random difference equation",
  "authors": [
    "Pawel Hitczenko"
  ],
  "comment": "to appear in Stochastic Processes and their Applications",
  "doi": "10.1016/j.spa.2011.06.007",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the extremes of a sequence of random variables $(R_n)$ defined by the recurrence $R_n=M_nR_{n-1}+q$, $n\\ge1$, where $R_0$ is arbitrary, $(M_n)$ are iid copies of a non--degenerate random variable $M$, $0\\le M\\le1$, and $q>0$ is a constant. We show that under mild and natural conditions on $M$ the suitably normalized extremes of $(R_n)$ converge in distribution to a double exponential random variable. This partially complements a result of de Haan, Resnick, Rootz\\'en, and de Vries who considered extremes of the sequence $(R_n)$ under the assumption that $\\P(M>1)>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-06-21T18:02:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "60F05"
    ],
    "keywords": [
      "random difference equation",
      "distribution",
      "convergence",
      "iid copies"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.4281H"
    }
  }
}