{
  "id": "0708.1676",
  "version": "v1",
  "published": "2007-08-13T09:48:40.000Z",
  "updated": "2007-08-13T09:48:40.000Z",
  "title": "Curve crossing for random walks reflected at their maximum",
  "authors": [
    "Ron Doney",
    "Ross Maller"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117906000000953 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2007, Vol. 35, No. 4, 1351-1373",
  "doi": "10.1214/009117906000000953",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $R_n=\\max_{0\\leq j\\leq n}S_j-S_n$ be a random walk $S_n$ reflected in its maximum. Except in the trivial case when $P(X\\ge0)=1$, $R_n$ will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend this by giving necessary and sufficient conditions for finiteness of passage times of $R_n$ above certain curved (power law) boundaries, as well. The intuition that a degree of heaviness of the negative tail of the distribution of the increments of $S_n$ is necessary for passage of $R_n$ above a high level is correct in most, but not all, cases, as we show. Conditions are also given for the finiteness of the expected passage time of $R_n$ above linear and square root boundaries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-08-13T09:48:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J15",
      "60F15",
      "60K05",
      "60G40",
      "60F05",
      "60G42"
    ],
    "keywords": [
      "random walks",
      "curve crossing",
      "square root boundaries",
      "horizontal boundary",
      "expected passage time"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.1676D"
    }
  }
}