{
  "id": "1407.2780",
  "version": "v2",
  "published": "2014-07-10T13:31:02.000Z",
  "updated": "2014-12-19T14:51:14.000Z",
  "title": "Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law",
  "authors": [
    "F. Götze",
    "A. N. Tikhomirov"
  ],
  "comment": "Abstract shortened. On p. 10 definition of $\\tilde{\\Lambda}^{(j)}_n$ has been changed, followed by appropriate changes in (6.29)-(6.58) etc. arXiv admin note: substantial text overlap with arXiv:1405.7820",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\mathbf X=(X_{jk})_{j,k=1}^n$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\\le j\\le k\\le n$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\\mathbf W=\\frac1{\\sqrt n}\\mathbf X$ to the semi-circular law assuming that $\\mathbf E X_{jk}=0$, $\\mathbf E X_{jk}^2=1$ and uniformly bounded eight moments. By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\\mathbf W$ and the semi--circular law is of order $O(n^{-1}\\log^{5}n)$ with high probability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-10T13:31:02.000Z",
      "abstract": "Let $\\mathbf X=(X_{jk})_{j,k=1}^n$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\\le j\\le k\\le n$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\\mathbf W=\\frac1{\\sqrt n}\\mathbf X$ to the semi-circular law assuming that $\\mathbf E X_{jk}=0$, $\\mathbf E X_{jk}^2=1$ and that \\begin{equation} \\sup_{n\\ge1}\\sup_{1\\le j,k\\le n}\\mathbf E|X_{jk}|^4=: \\mu_4<\\infty \\text{ and } \\sup_{1\\le j,k\\le n}|X_{jk}|\\le Dn^{\\frac14}. \\end{equation} By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\\mathbf W$ and the semi--circular law is of order $O(n^{-1}\\log^{5}n)$ with high probability.",
      "comment": "arXiv admin note: substantial text overlap with arXiv:1405.7820",
      "journal": null,
      "doi": null,
      "authors": [
        "F. Götze",
        "A. Tikhomirov"
      ]
    },
    {
      "version": "v2",
      "updated": "2014-12-19T14:51:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52"
    ],
    "keywords": [
      "empirical spectral distribution function",
      "convergence",
      "hermitian random matrix",
      "recursion argument",
      "kolmogorov distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2780G"
    }
  }
}