{
  "id": "1104.4953",
  "version": "v3",
  "published": "2011-04-26T15:57:36.000Z",
  "updated": "2012-05-03T08:06:06.000Z",
  "title": "A generalization of the Erdős-Turán law for the order of random permutation",
  "authors": [
    "Alexander Gnedin",
    "Alexander Iksanov",
    "Alexander Marynych"
  ],
  "comment": "21 pages, submitted to Combinatorics, Probability and Computing",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on $n$ integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erd\\H{o}s-Tur\\'an law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM$(\\theta)$ distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-03T08:06:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random permutation",
      "erdős-turán law",
      "generalization",
      "central limit theorem",
      "cycle structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.4953G"
    }
  }
}