{
  "id": "1012.0294",
  "version": "v3",
  "published": "2010-12-01T20:40:01.000Z",
  "updated": "2012-10-30T21:23:35.000Z",
  "title": "Sharp bounds on the rate of convergence of the empirical covariance matrix",
  "authors": [
    "Radosław Adamczak",
    "Alexander E. Litvak",
    "Alain Pajor",
    "Nicole Tomczak-Jaegermann"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "Let $X_1,..., X_N\\in\\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \\exp(-c\\sqrt{n}\\r)$ one has $ \\sup_{x\\in S^{n-1}} \\Big|\\frac{1/N}\\sum_{i=1}^N (|<X_i, x>|^2 - \\E|<X_i, x>|^2\\r)\\Big| \\leq C \\sqrt{\\frac{n/N}},$ where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $(<X_i,x>)_{i\\leq N, x\\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\\sqrt n)_{i\\leq N}$ exhibit uniformly a sub-exponential decay. As a consequence, if $A$ denotes the random matrix with columns $(X_i)$, then with overwhelming probability, the extremal singular values $\\lambda_{\\rm min}$ and $\\lambda_{\\rm max}$ of $AA^\\top$ satisfy the inequalities $ 1 - C\\sqrt{{n/N}} \\le {\\lambda_{\\rm min}/N} \\le \\frac{\\lambda_{\\rm max}/N} \\le 1 + C\\sqrt{{n/N}} $ which is a quantitative version of Bai-Yin theorem \\cite{BY} known for random matrices with i.i.d. entries.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-10-30T21:23:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "46B09",
      "52A21",
      "15A52",
      "60E15"
    ],
    "keywords": [
      "empirical covariance matrix",
      "sharp bounds",
      "convergence",
      "overwhelming probability",
      "independent centered random vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.0294A"
    }
  }
}