{
  "id": "2312.09652",
  "version": "v1",
  "published": "2023-12-15T09:56:13.000Z",
  "updated": "2023-12-15T09:56:13.000Z",
  "title": "The asymptotic distribution of the remainder in a certain base-$β$ expansion",
  "authors": [
    "Ira W. Herbst",
    "Jesper Møller",
    "Anne Marie Svane"
  ],
  "comment": "7 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X=\\sum_{k=1}^\\infty X_k \\beta^{-k}$ be the base-$\\beta$ expansion of a continuous random variable $X$ on the unit interval where $\\beta$ is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder $\\sum_{k=n+1}^\\infty X_k \\beta^{n-k}$ when $n$ tends to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-15T09:56:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F25",
      "62E17",
      "37A50"
    ],
    "keywords": [
      "asymptotic distribution",
      "unit interval",
      "golden ratio",
      "convergence rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}