{
  "id": "2406.17874",
  "version": "v1",
  "published": "2024-06-25T18:22:09.000Z",
  "updated": "2024-06-25T18:22:09.000Z",
  "title": "Central limits from generating functions",
  "authors": [
    "Mitchell Lee"
  ],
  "comment": "6 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(Y_n)_n$ be a sequence of $\\mathbb{R}^d$-valued random variables. Suppose that the generating function \\[f(x, z) = \\sum_{n = 0}^\\infty \\varphi_{Y_n}(x) z^n,\\] where $\\varphi_{Y_n}$ is the characteristic function of $Y_n$, extends to a function on a neighborhood of $\\{0\\} \\times \\{z : |z| \\leq 1\\} \\subset \\mathbb{R}^d \\times \\mathbb{C}$ which is meromorphic in $z$ and has no zeroes. We prove that if $1 / f(x, z)$ is twice differentiable, then there exists a constant $\\mu$ such that the distribution of $(Y_n - \\mu n) / \\sqrt{n}$ converges weakly to a normal distribution as $n \\to \\infty$. If $Y_n = X_1 + \\cdots + X_n$, where $(X_n)_n$ are i.i.d. random variables, then we recover the classical (Lindeberg$\\unicode{x2013}$L\\'evy) central limit theorem. We also prove the 2020 conjecture of Defant that if $\\pi_n \\in \\mathfrak{S}_n$ is a uniformly random permutation, then the distribution of $(\\operatorname{des} (s(\\pi_n)) + 1 - (3 - e) n) / \\sqrt{n}$ converges, as $n \\to \\infty$, to a normal distribution with variance $2 + 2e - e^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-25T18:22:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05"
    ],
    "keywords": [
      "generating function",
      "normal distribution",
      "central limit theorem",
      "characteristic function",
      "valued random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}