{
  "id": "1910.12701",
  "version": "v1",
  "published": "2019-10-28T14:20:47.000Z",
  "updated": "2019-10-28T14:20:47.000Z",
  "title": "Limiting behavior of largest entry of random tensor constructed by high-dimensional data",
  "authors": [
    "Tiefeng Jiang",
    "Junshan Xie"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let ${X}_{k}=(x_{k1}, \\cdots, x_{kp})', k=1,\\cdots,n$, be a random sample of size $n$ coming from a $p$-dimensional population. For a fixed integer $m\\geq 2$, consider a hypercubic random tensor $\\mathbf{{T}}$ of $m$-th order and rank $n$ with \\begin{eqnarray*} \\mathbf{{T}}= \\sum_{k=1}^{n}\\underbrace{{X}_{k}\\otimes\\cdots\\otimes {X}_{k}}_{m~multiple}=\\Big(\\sum_{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\\cdots x_{ki_{m}}\\Big)_{1\\leq i_{1},\\cdots, i_{m}\\leq p}. \\end{eqnarray*} Let $W_n$ be the largest off-diagonal entry of $\\mathbf{{T}}$. We derive the asymptotic distribution of $W_n$ under a suitable normalization for two cases. They are the ultra-high dimension case with $p\\to\\infty$ and $\\log p=o(n^{\\beta})$ and the high-dimension case with $p\\to \\infty$ and $p=O(n^{\\alpha})$ where $\\alpha,\\beta>0$. The normalizing constant of $W_n$ depends on $m$ and the limiting distribution of $W_n$ is a Gumbel-type distribution involved with parameter $m$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-28T14:20:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "62H10"
    ],
    "keywords": [
      "largest entry",
      "high-dimensional data",
      "limiting behavior",
      "hypercubic random tensor",
      "ultra-high dimension case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}