{
  "id": "0710.3262",
  "version": "v1",
  "published": "2007-10-17T10:42:05.000Z",
  "updated": "2007-10-17T10:42:05.000Z",
  "title": "$L^1$ bounds in normal approximation",
  "authors": [
    "Larry Goldstein"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/009117906000001123 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2007, Vol. 35, No. 5, 1888-1930",
  "doi": "10.1214/009117906000001123",
  "categories": [
    "math.PR"
  ],
  "abstract": "The zero bias distribution $W^*$ of $W$, defined though the characterizing equation $\\mathit{EW}f(W)=\\sigma^2Ef'(W^*)$ for all smooth functions $f$, exists for all $W$ with mean zero and finite variance $\\sigma^2$. For $W$ and $W^*$ defined on the same probability space, the $L^1$ distance between $F$, the distribution function of $W$ with $\\mathit{EW}=0$ and $Var(W)=1$, and the cumulative standard normal $\\Phi$ has the simple upper bound \\[\\Vert F-\\Phi\\Vert_1\\le2E|W^*-W|.\\] This inequality is used to provide explicit $L^1$ bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere $S(\\ell_n^p)$, simple random sampling and combinatorial central limit theorems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-17T10:42:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60F25",
      "60D05",
      "60C05"
    ],
    "keywords": [
      "normal approximation",
      "combinatorial central limit theorems",
      "zero bias distribution",
      "simple upper bound",
      "smooth functions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.3262G"
    }
  }
}