{
  "id": "2102.07699",
  "version": "v1",
  "published": "2021-02-15T17:52:25.000Z",
  "updated": "2021-02-15T17:52:25.000Z",
  "title": "Anti-concentration of random variables from zero-free regions",
  "authors": [
    "Marcus Michelen",
    "Julian Sahasrabudhe"
  ],
  "comment": "25 pages",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let $X$ be a random variable taking values in $\\{0,\\ldots,n\\}$ with $\\mathbb{P}(X = 0)\\mathbb{P}(X = n) > 0$ and with probability generating function $f_X$. We show that if all of the zeros $\\zeta$ of $f_X$ satisfy $|\\arg(\\zeta)| \\geq \\delta$ and $R^{-1} \\leq |\\zeta| \\leq R$ then \\[ \\operatorname{Var}(X) \\geq c R^{-2\\pi/\\delta}n, \\] where $c > 0$ is a absolute constant. We show that this result is sharp, up to the factor $2$ in the exponent of $R$. As a consequence, we are able to deduce a Littlewood--Offord type theorem for random variables that are not necessarily sums of i.i.d.\\ random variables.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-15T17:52:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random variable",
      "zero-free regions",
      "probability generating function",
      "anti-concentration",
      "littlewood-offord type theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}