{
  "id": "2308.14630",
  "version": "v1",
  "published": "2023-08-28T14:54:15.000Z",
  "updated": "2023-08-28T14:54:15.000Z",
  "title": "A fixed-point equation approach for the superdiffusive elephant random walk",
  "authors": [
    "Hélène Guérin",
    "Lucile Laulin",
    "Kilian Raschel"
  ],
  "comment": "38 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the elephant random walk in arbitrary dimension $d\\geq 1$. Our main focus is the limiting random variable appearing in the superdiffusive regime. Building on a link between the elephant random walk and P\\'olya-type urn models, we prove a fixed-point equation (or system in dimension two and larger) for the limiting variable. Based on this, we deduce several properties of the limit distribution, such as the existence of a density with support on $\\mathbb R^d$ for $d\\in\\{1,2,3\\}$, and we bring evidence for a similar result for $d\\geq 4$. We also investigate the moment-generating function of the limit and give, in dimension $1$, a non-linear recurrence relation for the moments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-28T14:54:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05",
      "60E10",
      "60J10",
      "60G50"
    ],
    "keywords": [
      "superdiffusive elephant random walk",
      "fixed-point equation approach",
      "non-linear recurrence relation",
      "polya-type urn models",
      "main focus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}