{
  "id": "0708.4408",
  "version": "v2",
  "published": "2007-08-31T18:50:27.000Z",
  "updated": "2008-05-07T13:10:22.000Z",
  "title": "Moments and distribution of the local times of a transient random walk on $\\Z^d$",
  "authors": [
    "Mathias Becker",
    "Wolfgang Konig"
  ],
  "comment": "9 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider an arbitrary transient random walk on $\\Z^d$ with $d\\in\\N$. Pick $\\alpha\\in[0,\\infty)$ and let $L_n(\\alpha)$ be the spatial sum of the $\\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $\\alpha$, $L_n(\\alpha)$ is the number of the $\\alpha$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(\\alpha)$ as $n\\to\\infty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by \\v{C}ern\\'y \\cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-05-07T13:10:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60J55",
      "60F15"
    ],
    "keywords": [
      "arbitrary transient random walk",
      "distribution",
      "step local times",
      "recurrent walks",
      "fold self-intersections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.4408B"
    }
  }
}