{
  "id": "math/0609682",
  "version": "v1",
  "published": "2006-09-25T09:42:34.000Z",
  "updated": "2006-09-25T09:42:34.000Z",
  "title": "On the second moment of the number of crossings by a stationary Gaussian process",
  "authors": [
    "Marie F. Kratz",
    "José R. León"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117906000000142 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 4, 1601-1607",
  "doi": "10.1214/009117906000000142",
  "categories": [
    "math.PR"
  ],
  "abstract": "Cram\\'{e}r and Leadbetter introduced in 1967 the sufficient condition \\[\\frac{r''(s)-r''(0)}{s}\\in L^1([0,\\delta],dx),\\qquad \\delta>0,\\] to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function $r$. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is still sufficient and necessary to have a finite variance of the number of any fixed level crossings. For the generalization to the number of a curve crossings, a condition on the curve has to be added to the Geman condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-25T09:42:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G10",
      "60G70"
    ],
    "keywords": [
      "second moment",
      "geman condition",
      "finite variance",
      "centered stationary gaussian process",
      "twice differentiable covariance function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9682K"
    }
  }
}