{
  "id": "2101.04052",
  "version": "v1",
  "published": "2021-01-11T17:37:47.000Z",
  "updated": "2021-01-11T17:37:47.000Z",
  "title": "An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process",
  "authors": [
    "Eran Assaf",
    "Jeremiah Buckley",
    "Naomi Feldheim"
  ],
  "comment": "35 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.CA"
  ],
  "abstract": "We study the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description of the variance of this random variable, under mild mixing conditions. In particular, we give a linear lower bound for any non-degenerate process. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth. Our results allow us to analyse a large number of interesting examples. We state some conjectures which generalise our results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-11T17:37:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G10",
      "60G15",
      "05A19",
      "37A46",
      "42A38"
    ],
    "keywords": [
      "stationary gaussian process",
      "asymptotic formula",
      "linear lower bound",
      "simple asymptotic description",
      "large number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}