{
  "id": "1404.5505",
  "version": "v1",
  "published": "2014-04-22T14:06:18.000Z",
  "updated": "2014-04-22T14:06:18.000Z",
  "title": "Pickands' constant $H_α$ does not equal $1/Γ(1/α)$, for small $α$",
  "authors": [
    "Adam J. Harper"
  ],
  "comment": "19 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Pickands' constants $H_{\\alpha}$ appear in various classical limit results about tail probabilities of suprema of Gaussian processes. It is an often quoted conjecture that perhaps $H_{\\alpha} = 1/\\Gamma(1/\\alpha)$ for all $0 < \\alpha \\leq 2$, but it is also frequently observed that this doesn't seem compatible with evidence coming from simulations. We prove the conjecture is false for small $\\alpha$, and in fact that $H_{\\alpha} \\geq (1.1527)^{1/\\alpha}/\\Gamma(1/\\alpha)$ for all sufficiently small $\\alpha$. The proof is a refinement of the \"conditioning and comparison\" approach to lower bounds for upper tail probabilities, developed in a previous paper of the author. Some calculations of hitting probabilities for Brownian motion are also involved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-22T14:06:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upper tail probabilities",
      "gaussian processes",
      "lower bounds",
      "classical limit results",
      "brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.5505H"
    }
  }
}