{
  "id": "1301.4059",
  "version": "v1",
  "published": "2013-01-17T11:48:33.000Z",
  "updated": "2013-01-17T11:48:33.000Z",
  "title": "Convex hulls of planar random walks with drift",
  "authors": [
    "Andrew R. Wade",
    "Chang Xu"
  ],
  "comment": "13 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Denote by $L_n$ the length of the perimeter of the convex hull of $n$ steps of a planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $n^{-1} L_n$ converges almost surely to a deterministic limit, and proved an upper bound on the variance $Var [ L_n] = O(n)$. We show that $n^{-1} Var [L_n]$ converges and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for $L_n$ in the non-degenerate case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-17T11:48:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60D05",
      "60J10",
      "60F05"
    ],
    "keywords": [
      "planar random walk",
      "convex hull",
      "central limit theorem",
      "finite second moment",
      "degenerate class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}