{
  "id": "2012.03758",
  "version": "v1",
  "published": "2020-12-07T14:52:09.000Z",
  "updated": "2020-12-07T14:52:09.000Z",
  "title": "Uniform Central Limit Theorem for self normalized sums in high dimensions",
  "authors": [
    "Debraj Das"
  ],
  "comment": "25 pages",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "In this article, we are interested in the normal approximation of the self-normalized random vector $\\Big(\\frac{\\sum_{i=1}^{n}X_{i1}}{\\sqrt{\\sum_{i=1}^{n}X_{i1}^2}},\\dots,\\frac{\\sum_{i=1}^{n}X_{ip}}{\\sqrt{\\sum_{i=1}^{n}X_{ip}^2}}\\Big)$ in $\\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\\mathcal{A}^{re}=\\{\\prod_{j=1}^{p}[a_j,b_j]\\cap\\mathcal{R}:-\\infty\\leq a_j\\leq b_j\\leq \\infty, j=1,\\ldots,p\\}$, where $X_1,\\dots,X_n$ are non-degenerate independent $p-$dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate of $\\log p$ in the uniform central limit theorem (UCLT) under variety of moment conditions. When $X_{ij}$'s have $(2+\\delta)$th absolute moment for some $0< \\delta\\leq 1$, the optimal rate of $\\log p$ is $o\\big(n^{\\delta/(2+\\delta)}\\big)$. When $X_{ij}$'s are independent and identically distributed (iid) across $(i,j)$, even $(2+\\delta)$th absolute moment of $X_{11}$ is not needed. Only under the condition that $X_{11}$ is in the domain of attraction of the normal distribution, the growth rate of $\\log p$ can be made to be $o(\\eta_n)$ for some $\\eta_n\\rightarrow 0$ as $n\\rightarrow \\infty$. We also establish that the rate of $\\log p$ can be pushed to $\\log p =o(n^{1/2})$ if we assume the existence of fourth moment of $X_{ij}$'s. By an example, it is shown however that the rate of growth of $\\log p$ can not further be improved from $n^{1/2}$ as a power of $n$. As an application, we found respective versions of the high dimensional UCLT for component-wise Student's t-statistic. An important aspect of the these UCLTs is that it does not require the existence of some exponential moments even when dimension $p$ grows exponentially with some power of $n$, as opposed to the UCLT of normalized sums. Only the existence of some absolute moment of order $\\in [2,4]$ is sufficient.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-07T14:52:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60B12",
      "62E20"
    ],
    "keywords": [
      "uniform central limit theorem",
      "self normalized sums",
      "high dimensions",
      "th absolute moment",
      "random vector"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}