{
  "id": "math/0406392",
  "version": "v3",
  "published": "2004-06-20T11:33:40.000Z",
  "updated": "2006-03-10T16:31:23.000Z",
  "title": "Level Crossing Probabilities I: One-dimensional Random Walks and Symmetrization",
  "authors": [
    "Rainer Siegmund-Schultze",
    "Heinrich von Weizsaecker"
  ],
  "comment": "10 pages, some references added, typos corrected",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing symmetry the (sharp) inequality P(|X+Y| <= 1) < 2 P(|X-Y| <= 1) for independent identically distributed X,Y is used. In part II we shall discuss the connection of this result to 'polygonal recurrence' of higher-dimensional walks and some conjectures on directionally random walks in the sense of Mauldin, Monticino and v.Weizsaecker [5].",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-03-10T16:31:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51"
    ],
    "keywords": [
      "level crossing probabilities",
      "probability",
      "arbitrary one-dimensional random walk",
      "symmetrization",
      "independent increments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6392S"
    }
  }
}