{
  "id": "math/0503475",
  "version": "v1",
  "published": "2005-03-23T07:30:54.000Z",
  "updated": "2005-03-23T07:30:54.000Z",
  "title": "On the distribution of the maximum of a gaussian field with d parameters",
  "authors": [
    "Jean-Marc Azais",
    "Mario Wschebor"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/105051604000000602 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2005, Vol. 15, No. 1A, 254-278",
  "doi": "10.1214/105051604000000602",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let I be a compact d-dimensional manifold, let X:I\\to R be a Gaussian process with regular paths and let F_I(u), u\\in R, be the probability distribution function of sup_{t\\in I}X(t). We prove that under certain regularity and nondegeneracy conditions, F_I is a C^1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u\\to +\\infty. This is a partial extension of previous results by the authors in the case d=1. Our methods use strongly the so-called Rice formulae for the moments of the number of roots of an equation of the form Z(t)=x, where Z:I\\to R^d is a random field and x is a fixed point in R^d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-03-23T07:30:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G70"
    ],
    "keywords": [
      "gaussian field",
      "parameters",
      "compact d-dimensional manifold",
      "probability distribution function",
      "rice formulae"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......3475A"
    }
  }
}