{
  "id": "2307.10027",
  "version": "v1",
  "published": "2023-07-19T15:12:05.000Z",
  "updated": "2023-07-19T15:12:05.000Z",
  "title": "Iterated-logarithm laws for convex hulls of random walks with drift",
  "authors": [
    "Wojciech Cygan",
    "Nikola Sandrić",
    "Stjepan Šebek",
    "Andrew Wade"
  ],
  "comment": "27 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero--one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-19T15:12:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60D05",
      "60F15",
      "60J65",
      "52A22"
    ],
    "keywords": [
      "convex hull",
      "iterated-logarithm law",
      "strassens functional law",
      "multidimensional random walks",
      "planar random walk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}