{
  "id": "1707.04130",
  "version": "v1",
  "published": "2017-07-06T21:46:36.000Z",
  "updated": "2017-07-06T21:46:36.000Z",
  "title": "A martingale approach for the elephant random walk",
  "authors": [
    "Bernard Bercu"
  ],
  "categories": [
    "math.PR",
    "physics.data-an"
  ],
  "abstract": "The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter $p$ which lies between zero and one. This behavior is totally different in the diffusive regime $0 \\leq p <3/4$, the critical regime $p=3/4$, and the superdiffusive regime $3/4<p \\leq 1$. Notwithstanding of this trichotomy, we provide some new results on the almost sure convergence and the asymptotic normality of the ERW.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-06T21:46:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "martingale approach",
      "one-dimensional elephant random walk",
      "asymptotic behavior",
      "memory parameter",
      "sure convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}