{
  "id": "1804.07696",
  "version": "v1",
  "published": "2018-04-20T15:56:00.000Z",
  "updated": "2018-04-20T15:56:00.000Z",
  "title": "Central limit theorems from the roots of probability generating functions",
  "authors": [
    "Marcus Michelen",
    "Julian Sahasrabudhe"
  ],
  "categories": [
    "math.PR",
    "math.CA",
    "math.CO"
  ],
  "abstract": "For each $n$, let $X_n \\in \\{0,\\ldots,n\\}$ be a random variable with mean $\\mu_n$, standard deviation $\\sigma_n$, and let \\[ P_n(z) = \\sum_{k=0}^n \\mathbb{P}( X_n = k) z^k ,\\] be its probability generating function. We show that if none of the complex zeros of the polynomials $\\{ P_n(z)\\}$ are contained in a neighbourhood of $1 \\in \\mathbb{C}$ and $\\sigma_n > n^{\\varepsilon}$ for some $\\varepsilon >0$, then $ X_n^* =(X_n - \\mu_n)\\sigma^{-1}_n$ tends to a normal random variable $Z \\sim \\mathcal{N}(0,1)$ in distribution as $n \\rightarrow \\infty$. Moreover, we show this result is sharp in the sense that there exist sequences of random variables $\\{X_n\\}$ with $\\sigma_n > C\\log n $ for which $P_n(z)$ has no roots near $1$ and $X_n^*$ is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of $P_n(z)$ and the distribution of the random variables $X_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-20T15:56:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "26C05",
      "26C10"
    ],
    "keywords": [
      "probability generating function",
      "central limit theorems",
      "random variable",
      "results disprove",
      "complex zeros"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}