{
  "id": "math/0104113",
  "version": "v2",
  "published": "2001-04-10T17:47:39.000Z",
  "updated": "2001-06-25T20:11:08.000Z",
  "title": "A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices",
  "authors": [
    "Alexander Soshnikov"
  ],
  "comment": "This is a preliminary version. Minor misprints are corrected",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix $ X^* \\* X (X^t \\*X) $ converges to the Tracy-Widom law as $ n, p $ (the dimensions of $ X $) tend to $ \\infty $ in some ratio $ n/p \\to \\gamma>0. $ We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy-Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner matrices allows to extend the results by Johansson and Johnstone to the case of $ X $ with non-Gaussian entries, provided $ n-p =O(p^{1/3}) . $ We also prove that $ \\lambda_{max} \\leq (n^{1/2}+p^{1/2})^2 +O(p^{1/2}\\*\\log(p)) $ (a.e.) for general $ \\gamma >0.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2001-06-25T20:11:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sample covariance matrices",
      "largest eigenvalues",
      "distribution",
      "universality",
      "largest principal component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......4113S"
    }
  }
}