{
  "id": "1707.06905",
  "version": "v1",
  "published": "2017-07-21T14:04:05.000Z",
  "updated": "2017-07-21T14:04:05.000Z",
  "title": "A strong invariance principle for the elephant random walk",
  "authors": [
    "Cristian F. Coletti",
    "Renato Gava",
    "Gunter M. Schütz"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a non-Markovian discrete-time random walk on $\\mathbb{Z}$ with unbounded memory called the elephant random walk (ERW). We prove a strong invariance principle for the ERW. More specifically, we prove that, under a suitable scaling and in the diffusive regime as well as at the critical value $p_c=3/4$ where the model is marginally superdiffusive, the ERW is almost surely well approximated by a Brownian motion. As a by-product of our result we get the law of iterated logarithm and the central limit theorem for the ERW.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-21T14:04:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elephant random walk",
      "strong invariance principle",
      "non-markovian discrete-time random walk",
      "central limit theorem",
      "brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}