{
  "id": "1701.03946",
  "version": "v1",
  "published": "2017-01-14T16:44:02.000Z",
  "updated": "2017-01-14T16:44:02.000Z",
  "title": "The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives",
  "authors": [
    "I-Shing Hu",
    "Chih-Chung Chang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $ p_n(x) $ be a random polynomial of degree $n$ and $\\{Z^{(n)}_j\\}_{j=1}^n$ and $\\{X^{n, k}_j\\}_{j=1}^{n-k}, k<n$, be the zeros of $p_n$ and $p_n^{(k)}$, the $k$th derivative of $p_n$, respectively. We show that if the linear statistics $\\frac{1}{a_n} \\left[ f\\left( \\frac {Z^{(n)}_1}{b_n} \\right)+ \\cdots + f \\left(\\frac {Z^{(n)}_n}{b_n} \\right) \\right]$ associated with $\\{Z^{(n)}_j\\}$ has a limit as $n\\to\\infty$ at some mode of convergence, the linear statistics associated with $\\{X^{n, k}_j\\}$ converges to the same limit at the same mode. Similar statement also holds for the centered linear statistics associated with the zeros of $p_n$ and $p_n^{(k)}$, provided the zeros $\\{Z^{(n)}_j\\}$ and the sequences $\\{a_n\\}$ and $\\{b_n\\}$ of positive numbers satisfy some mild conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-14T16:44:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random polynomial",
      "common limit",
      "derivative",
      "similar statement",
      "centered linear statistics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}