{
  "id": "2103.05402",
  "version": "v1",
  "published": "2021-03-09T12:53:12.000Z",
  "updated": "2021-03-09T12:53:12.000Z",
  "title": "Quantitative CLT for linear eigenvalue statistics of Wigner matrices",
  "authors": [
    "Zhigang Bao",
    "Yukun He"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions $f\\in C^5(\\mathbb R)$, we show that the convergence rate is either $N^{-1/2+\\varepsilon}$ or $N^{-1+\\varepsilon}$, depending on the first Chebyshev coefficient of $f$ and the third moment of the diagonal matrix entries. The condition that distinguishes these two rates is necessary and sufficient. For a general class of test functions, we further identify matching lower bounds for the convergence rates. In addition, we identify an explicit, non-universal contribution in the linear eigenvalue statistics, which is responsible for the slow rate $N^{-1/2+\\varepsilon}$ for non-Gaussian ensembles. By removing this non-universal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate $N^{-1+\\varepsilon}$ for all test functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-09T12:53:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "60F05"
    ],
    "keywords": [
      "wigner matrices",
      "quantitative clt",
      "test functions",
      "near-optimal convergence rate",
      "first chebyshev coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}