{
  "id": "1412.7444",
  "version": "v1",
  "published": "2014-12-23T16:58:36.000Z",
  "updated": "2014-12-23T16:58:36.000Z",
  "title": "Max-stable processes and stationary systems of Lévy particles",
  "authors": [
    "Sebastian Engelke",
    "Zakhar Kabluchko"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study stationary max-stable processes $\\{\\eta(t)\\colon t\\in\\mathbb R\\}$ admitting a representation of the form $\\eta(t)=\\max_{i\\in\\mathbb N}(U_i+ Y_i(t))$, where $\\sum_{i=1}^{\\infty} \\delta_{U_i}$ is a Poisson point process on $\\mathbb R$ with intensity ${\\rm e}^{-u} {\\rm d} u$, and $Y_1,Y_2,\\ldots$ are i.i.d.\\ copies of a process $\\{Y(t)\\colon t\\in\\mathbb R\\}$ obtained by running a L\\'evy process for positive $t$ and a dual L\\'evy process for negative $t$. We give a general construction of such processes, where the restrictions of $Y$ to the positive and negative half-axes are L\\'evy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form $M_n(t)=\\max_{i=1,\\ldots,n} \\xi_i(s_n+t)$, where $\\xi_1,\\xi_2,\\ldots$ are i.i.d.\\ L\\'evy processes and $s_n$ is a sequence such that $s_n\\sim c \\log n$ with $c>0$. Also, we consider maxima of the form $\\max_{i=1,\\ldots,n} Z_i(t/\\log n)$, where $Z_1,Z_2,\\ldots$ are i.i.d.\\ Ornstein--Uhlenbeck processes driven by an $\\alpha$-stable noise with skewness parameter $\\beta=-1$. After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick [Extreme values of independent stochastic processes, \\textit{J.\\ Appl.\\ Probab.}, 14 (1977), pp.\\ 732--739] to the totally skewed $\\alpha$-stable case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-23T16:58:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "60G51",
      "60G10",
      "60G55"
    ],
    "keywords": [
      "stationary systems",
      "lévy particles",
      "levy processes",
      "dual levy process",
      "independent stochastic processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}