{
  "id": "1708.04470",
  "version": "v1",
  "published": "2017-08-15T12:08:28.000Z",
  "updated": "2017-08-15T12:08:28.000Z",
  "title": "On the centre of mass of a random walk",
  "authors": [
    "Chak Hei Lo",
    "Andrew R. Wade"
  ],
  "comment": "25 pages, 1 colour figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "For a random walk $S_n$ on $\\mathbb{R}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n = n^{-1} \\sum_{i=1}^n S_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, $G_n$ is recurrent if $d=1$ and transient if $d \\geq 2$. In the transient case we show that $G_n$ has diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which $G_n$ is transient in $d=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-15T12:08:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F05",
      "60J10"
    ],
    "keywords": [
      "simple symmetric random walk",
      "results extend work",
      "local limit theorem",
      "finite second moment",
      "lattice distributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}