{
  "id": "1109.0611",
  "version": "v3",
  "published": "2011-09-03T09:47:20.000Z",
  "updated": "2013-10-28T16:09:27.000Z",
  "title": "On the rate of convergence to the semi-circular law",
  "authors": [
    "Friedrich Götze",
    "Alexander Tikhomirov"
  ],
  "comment": "This version fills a gap in the previous version using martingale large deviation bounds. Here iterative expansions are used in Section 4, Proposition 4.1 with supporting bounds in Section 3, Lemmas 3.2--3.9, together with expansion of resolvents in rows and columns and re-expressing some terms again in the original resolvents (e.g. (3.45)) and Lemmas 3.10--3.19",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\mathbf X=(X_{jk})$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\\le j\\le k$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\\mathbf X$ to the semi-circular law assuming that $\\mathbf E X_{jk}=0$, $\\mathbf E X_{jk}^2=1$ and that the distributions of the matrix elements $X_{jk}$ have a uniform sub exponential decay in the sense that there exists a constant $\\varkappa>0$ such that for any $1\\le j\\le k\\le n$ and any $t\\ge 1$ we have $$ \\Pr\\{|X_{jk}|>t\\}\\le \\varkappa^{-1}\\exp\\{-t^{\\varkappa}\\}. $$ By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\\mathbf W=\\frac1{\\sqrt n}\\mathbf X$ and the semicircular law is of order $O(n^{-1}\\log^b n)$ with some positive constant $b>0$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-10-28T16:09:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52"
    ],
    "keywords": [
      "semi-circular law",
      "convergence",
      "uniform sub exponential decay",
      "hermitian random matrix",
      "empirical spectral distribution function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.0611G"
    }
  }
}