{
  "id": "1108.3886",
  "version": "v1",
  "published": "2011-08-19T03:51:41.000Z",
  "updated": "2011-08-19T03:51:41.000Z",
  "title": "On generic chaining and the smallest singular value of random matrices with heavy tails",
  "authors": [
    "Shahar Mendelson",
    "Grigoris Paouris"
  ],
  "comment": "42 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We present a very general chaining method which allows one to control the supremum of the empirical process $\\sup_{h \\in H} |N^{-1}\\sum_{i=1}^N h^2(X_i)-\\E h^2|$ in rather general situations. We use this method to establish two main results. First, a quantitative (non asymptotic) version of the classical Bai-Yin Theorem on the singular values of a random matrix with i.i.d entries that have heavy tails, and second, a sharp estimate on the quadratic empirical process when $H=\\{\\inr{t,\\cdot} : t \\in T\\}$, $T \\subset \\R^n$ and $\\mu$ is an isotropic, unconditional, log-concave measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-19T03:51:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smallest singular value",
      "heavy tails",
      "random matrices",
      "generic chaining",
      "general chaining method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.3886M"
    }
  }
}