{
  "id": "2208.09928",
  "version": "v1",
  "published": "2022-08-21T17:30:10.000Z",
  "updated": "2022-08-21T17:30:10.000Z",
  "title": "Variations of Central Limit Theorems and Stirling numbers of the First Kind",
  "authors": [
    "Bernhard Heim",
    "Markus Neuhauser"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We construct a new parametrization of double sequences $\\{A_{n,k}(s)\\}_{n,k}$ between $A_{n,k}(0)= \\binom{n-1}{k-1}$ and $A_{n,k}(1)= \\frac{1}{n!}\\stirl{n}{k}$, where $\\stirl{n}{k}$ are the unsigned Stirling numbers of the first kind. For each $s$ we prove a central limit theorem and a local limit theorem. This extends the de\\,Moivre--Laplace central limit theorem and Goncharov's result, that unsigned Stirling numbers of the first kind are asymptotically normal. Herewith, we provide several applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-21T17:30:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "60F05"
    ],
    "keywords": [
      "central limit theorem",
      "first kind",
      "unsigned stirling numbers",
      "variations",
      "local limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}