{
  "id": "1111.3159",
  "version": "v5",
  "published": "2011-11-14T09:55:29.000Z",
  "updated": "2015-04-13T05:34:13.000Z",
  "title": "On the error bound in a combinatorial central limit theorem",
  "authors": [
    "Louis H. Y. Chen",
    "Xiao Fang"
  ],
  "comment": "Published at http://dx.doi.org/10.3150/13-BEJ569 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)",
  "journal": "Bernoulli 2015, Vol. 21, No. 1, 335-359",
  "doi": "10.3150/13-BEJ569",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let $\\mathbb{X}=\\{X_{ij}: 1\\le i,j\\le n\\}$ be an $n\\times n$ array of independent random variables where $n\\ge2$. Let $\\pi$ be a uniform random permutation of $\\{1,2,\\dots,n\\}$, independent of $\\mathbb{X}$, and let $W=\\sum_{i=1}^nX_{i\\pi(i)}$. Suppose $\\mathbb{X}$ is standardized so that ${\\mathbb{E}}W=0,\\operatorname {Var}(W)=1$. We prove that the Kolmogorov distance between the distribution of $W$ and the standard normal distribution is bounded by $451\\sum_{i,j=1}^n{\\mathbb{E}}|X_{ij}|^3/n$. Our approach is by Stein's method of exchangeable pairs and the use of a concentration inequality.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-09-26T22:25:28.000Z",
      "title": "On the Error Bound in a Combinatorial Central Limit Theorem",
      "abstract": "Let $\\mathbb{X}={X_{ij}: 1\\le i,j \\le n}$ be an $n\\times n$ array of independent random variables where $n \\ge 2$. Let $\\pi$ be a uniform random permutation of ${1,2,..., n}$, independent of $\\mathbb{X}$, and let $W=\\sum_{i=1}^n X_{i\\pi(i)}$. Suppose $\\mathbb{X}$ is standardized so that $\\E W=0, \\Var(W)=1$. We prove that the Kolmogorov distance between the distribution of $W$ and the standard normal distribution is bounded by $451\\sum_{i,j=1}^n \\E |X_{ij}|^3/n$. Our approach is by Stein's method of exchangeable pairs and the use of a concentration inequality.",
      "comment": "27 page; final version submitted to the publisher",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-04-13T05:34:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "62E20",
      "62G99"
    ],
    "keywords": [
      "combinatorial central limit theorem",
      "error bound",
      "standard normal distribution",
      "uniform random permutation",
      "independent random variables"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.3159C"
    }
  }
}