{
  "id": "1704.05205",
  "version": "v1",
  "published": "2017-04-18T05:23:03.000Z",
  "updated": "2017-04-18T05:23:03.000Z",
  "title": "Distances between Random Orthogonal Matrices and Independent Normals",
  "authors": [
    "Tiefeng Jiang",
    "Yutao Ma"
  ],
  "comment": "42 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\Gamma_n$ be an $n\\times n$ Haar-invariant orthogonal matrix. Let $ Z_n$ be the $p\\times q$ upper-left submatrix of $\\Gamma_n,$ where $p=p_n$ and $q=q_n$ are two positive integers. Let $G_n$ be a $p\\times q$ matrix whose $pq$ entries are independent standard normals. In this paper we consider the distance between $\\sqrt{n} Z_n$ and $ G_n$ in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. We prove that each of the first three distances goes to zero as long as $pq/n$ goes to zero, and not so this rate is sharp in the sense that each distance does not go to zero if $(p, q)$ sits on the curve $pq=\\sigma n$, where $\\sigma$ is a constant. However, it is different for the Euclidean distance, which goes to zero provided $pq^2/n$ goes to zero, and not so if $(p,q)$ sits on the curve $pq^2=\\sigma n.$ A previous work by Jiang \\cite{Jiang06} shows that the total variation distance goes to zero if both $p/\\sqrt{n}$ and $q/\\sqrt{n}$ go to zero, and it is not true provided $p=c\\sqrt{n}$ and $q=d\\sqrt{n}$ with $c$ and $d$ being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as $pq/n\\to 0$ and the distance does not go to zero if $pq=\\sigma n$ for some constant $\\sigma$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-18T05:23:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "28C10",
      "51F25",
      "60B15",
      "62E17"
    ],
    "keywords": [
      "random orthogonal matrices",
      "total variation distance",
      "independent normals",
      "euclidean distance",
      "independent standard normals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}