{
  "id": "2401.09418",
  "version": "v1",
  "published": "2024-01-17T18:57:49.000Z",
  "updated": "2024-01-17T18:57:49.000Z",
  "title": "Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions",
  "authors": [
    "Benedikt Rednoß",
    "Christoph Thäle"
  ],
  "comment": "16 pages, 2 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "This paper deals with sequences of random variables $X_n$ only taking values in $\\{0,\\ldots,n\\}$. The probability generating functions of such random variables are polynomials of degree $n$. Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for $X_n$ is established in a unified way. In the real rooted case the result is classical and only involves the variances of $X_n$, while in the cyclotomic case the fourth cumulants or moments of $X_n$ appear in addition. The proofs are elementary and based on the Stein-Tikhomirov method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-17T18:57:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "60F05"
    ],
    "keywords": [
      "fourth moment theorem",
      "cyclotomic generating functions",
      "random variables",
      "quantification",
      "quantitative normal approximation bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}