{
  "id": "1912.05473",
  "version": "v1",
  "published": "2019-12-11T17:10:03.000Z",
  "updated": "2019-12-11T17:10:03.000Z",
  "title": "Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices",
  "authors": [
    "Haoyu Wang"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We prove the first explicit rate of convergence to the Tracy-Widom distribution for the fluctuation of the largest eigenvalue of sample covariance matrices that are not integrable. Our primary focus is matrices of type $ X^*X $ and the proof follows the Erd\\\"{o}s-Schlein-Yau dynamical method. We use a recent approach to the analysis of the Dyson Brownian motion from [5] to obtain a quantitative error estimate for the local relaxation flow at the edge. Together with a quantitative version of the Green function comparison theorem, this gives the rate of convergence. Combined with a result of Lee-Schnelli [26], some quantitative estimates also hold for more general separable sample covariance matrices $ X^* \\Sigma X $ with general diagonal population $ \\Sigma $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-11T17:10:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52"
    ],
    "keywords": [
      "largest eigenvalue",
      "quantitative universality",
      "general separable sample covariance matrices",
      "green function comparison theorem",
      "first explicit rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}