{
  "id": "1607.06873",
  "version": "v1",
  "published": "2016-07-23T01:19:38.000Z",
  "updated": "2016-07-23T01:19:38.000Z",
  "title": "A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices",
  "authors": [
    "Xiucai Ding",
    "Fan Yang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we prove a necessary and sufficient condition for the edge universality of covariance matrices, which is conjectured by Pillai and Yin. Consider the sample covariance matrices of the form $XX^{*}$, where $X$ is an $M \\times N$ rectangular matrix with i.i.d entries $X_{ij}=x_{ij}$ satisfying $\\mathbb{E} x_{ij}=0$ and $ \\mathbb{E} |x_{ij}|^2 ={N}^{-1}$. Under the assumption $\\lim_{N \\to \\infty} {N}/{M}=d \\in (0, \\infty)$, we prove that the Tracy-Widom law holds at the soft edge (i.e., for the largest eigenvalues) if and only if $\\lim_{s \\rightarrow \\infty}s^4 \\mathbb{P}(\\vert \\sqrt{N} x_{ij} \\vert \\geq s)=0$. This condition was first proposed for Wigner matrices by Lee and Yin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-23T01:19:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "largest singular values",
      "edge universality",
      "sufficient condition",
      "sample covariance matrices",
      "tracy-widom law holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}