{
  "id": "1404.1480",
  "version": "v1",
  "published": "2014-04-05T13:58:05.000Z",
  "updated": "2014-04-05T13:58:05.000Z",
  "title": "Weak convergence of partial maxima processes in the $M_{1}$ topology",
  "authors": [
    "Danijel Krizmanić"
  ],
  "comment": "17 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "It is known that for a sequence of independent and identically distributed random variables $(X_{n})$ the regular variation condition is equivalent to weak convergence of partial maxima $M_{n}= \\max\\{X_{1}, \\ldots, X_{n}\\}$, appropriately scaled. A functional version of this is known to be true as well, the limit process being an extremal process, and the convergence takes place in the space of c\\`{a}dl\\`{a}g functions endowed with the Skorohod $J_{1}$ topology. We first show that weak convergence of partial maxima $M_{n}$ holds also for a class of weakly dependent sequences under the joint regular variation condition. Then using this result we obtain a corresponding functional version for the processes of partial maxima $M_{n}(t) = \\bigvee_{i=1}^{\\lfloor nt \\rfloor}X_{i},\\,t \\in [0,1]$, but with respect to the Skorohod $M_{1}$ topology, which is weaker than the more usual $J_{1}$ topology. We also show that the $M_{1}$ convergence generally can not be replaced by the $J_{1}$ convergence. Applications of our main results to moving maxima, squared GARCH and ARMAX processes are also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-05T13:58:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak convergence",
      "partial maxima processes",
      "joint regular variation condition",
      "armax processes",
      "main results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.1480K"
    }
  }
}