{
  "id": "1007.4870",
  "version": "v3",
  "published": "2010-07-28T04:29:33.000Z",
  "updated": "2012-06-12T07:40:31.000Z",
  "title": "(Very) short proof of Rayleigh's Theorem (and extensions)",
  "authors": [
    "Olivier Bernardi"
  ],
  "comment": "2 pages",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Consider a walk in the plane made of $n$ unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its starting point is $1/(n+1)$. We give an elementary proof of this result. We also prove the following generalization valid for any probability distribution $\\mu$ on the positive real numbers: if two walkers start at the same point and make respectively $m$ and $n$ independent steps with uniformly random directions and with lengths chosen according to $\\mu$, then the probability that the first walker ends farther than the second is $m/(m+n)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-06-12T07:40:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "short proof",
      "extensions",
      "first walker ends farther",
      "rayleighs theorem asserts",
      "unit steps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 2,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.4870B"
    }
  }
}