{
  "id": "1404.0172",
  "version": "v2",
  "published": "2014-04-01T09:22:37.000Z",
  "updated": "2015-02-03T14:36:12.000Z",
  "title": "The correlation measures of finite sequences: limiting distributions and minimum values",
  "authors": [
    "Kai-Uwe Schmidt"
  ],
  "comment": "19 pages, this version contains small changes taking into account referee comments",
  "categories": [
    "math.PR",
    "math.CO",
    "math.NT"
  ],
  "abstract": "Three measures of pseudorandomness of finite binary sequences were introduced by Mauduit and S\\'ark\\\"ozy in 1997 and have been studied extensively since then: the normality measure, the well-distribution measure, and the correlation measure of order r. Our main result is that the correlation measure of order r for random binary sequences converges strongly, and so has a limiting distribution. This solves a problem due to Alon, Kohayakawa, Mauduit, Moreira, and R\\\"odl. We also show that the best known lower bounds for the minimum values of the correlation measures are simple consequences of a celebrated result due to Welch, concerning the maximum nontrivial scalar products over a set of vectors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-01T09:22:37.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-03T14:36:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K45",
      "60C05",
      "68R15"
    ],
    "keywords": [
      "correlation measure",
      "minimum values",
      "limiting distribution",
      "finite sequences",
      "maximum nontrivial scalar products"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.0172S"
    }
  }
}