{
  "id": "1609.07909",
  "version": "v1",
  "published": "2016-09-26T10:07:16.000Z",
  "updated": "2016-09-26T10:07:16.000Z",
  "title": "Pickands' constant at first order in an expansion around Brownian motion",
  "authors": [
    "Mathieu Delorme",
    "Alberto Rosso",
    "Kay Jörg Wiese"
  ],
  "comment": "13 pages, 3 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the theory of extreme values of Gaussian processes, many results are expressed in terms of the Pickands constant $\\mathcal{H}_{\\alpha}$. This constant depends on the local self-similarity exponent $\\alpha$ of the process, i.e. locally it is a fractional Brownian motion (fBm) of Hurst index $H=\\alpha/2$. Despite its importance, only two values of the Pickands constant are known: ${\\cal H}_1 =1$ and ${\\cal H}_2=1/\\sqrt{\\pi}$. Here, we extend the recent perturbative approach to fBm to include drift terms. This allows us to investigate the Pickands constant $\\mathcal{H}_{\\alpha}$ around standard Brownian motion ($\\alpha =1$) and to derive the new exact result $\\mathcal{H}_{\\alpha}=1 - (\\alpha-1) \\gamma_{\\rm E} + \\mathcal{O}\\!\\left( \\alpha-1\\right)^{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-26T10:07:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first order",
      "pickands constant",
      "standard brownian motion",
      "local self-similarity exponent",
      "fractional brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}