{
  "id": "0903.2323",
  "version": "v2",
  "published": "2009-03-13T07:09:30.000Z",
  "updated": "2009-08-27T16:22:41.000Z",
  "title": "Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles",
  "authors": [
    "Radosław Adamczak",
    "Alexander E. Litvak",
    "Alain Pajor",
    "Nicole Tomczak-Jaegermann"
  ],
  "comment": "Exposition changed, several explanatory remarks added, some proofs simplified",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "Let $K$ be an isotropic convex body in $\\R^n$. Given $\\eps>0$, how many independent points $X_i$ uniformly distributed on $K$ are needed for the empirical covariance matrix to approximate the identity up to $\\eps$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits. More precisely, let $X\\in\\R^n$ be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector $X$ is a random point in an isotropic convex body. We show that for any $\\eps>0$, there exists $C(\\eps)>0$, such that if $N\\sim C(\\eps) n$ and $(X_i)_{i\\le N}$ are i.i.d. copies of $X$, then $ \\Big\\|\\frac{1}{N}\\sum_{i=1}^N X_i\\otimes X_i - \\Id\\Big\\| \\le \\epsilon, $ with probability larger than $1-\\exp(-c\\sqrt n)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-08-27T16:22:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "46B09",
      "52A21",
      "15A52",
      "60E15"
    ],
    "keywords": [
      "empirical covariance matrix",
      "log-concave ensembles",
      "quantitative estimates",
      "isotropic convex body",
      "convergence"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1090/S0894-0347-09-00650-X",
      "journal": "Journal of the American Mathematical Society",
      "year": 2010,
      "month": "Apr",
      "volume": 23,
      "number": 2,
      "pages": 535
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JAMS...23..535A"
    }
  }
}