{
  "id": "1908.06611",
  "version": "v1",
  "published": "2019-08-19T06:39:47.000Z",
  "updated": "2019-08-19T06:39:47.000Z",
  "title": "Strong law of large numbers for a function of the local times of a transient random walk in $\\mathbb Z^d$",
  "authors": [
    "I. M. Asymont",
    "D. Korshunov"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "For an arbitrary transient random walk $(S_n)_{n\\ge 0}$ in $\\mathbb Z^d$, $d\\ge 1$, we prove a strong law of large numbers for the spatial sum $\\sum_{x\\in\\mathbb Z^d}f(l(n,x))$ of a function $f$ of the local times $l(n,x)=\\sum_{i=0}^n\\mathbb I\\{S_i=x\\}$. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and Erd\\H{o}s), which corresponds to a function $f(i)=\\mathbb I\\{i\\ge 1\\}$; (b) $\\alpha$-fold self-intersections of the random walk (studied by Becker and K\\\"{o}nig), which corresponds to $f(i)=i^\\alpha$; (c) sites visited by the random walk exactly $j$ times (considered by Erd\\H{o}s and Taylor and by Pitt), where $f(i)=\\mathbb I\\{i=j\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-19T06:39:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60J55",
      "60F15"
    ],
    "keywords": [
      "strong law",
      "local times",
      "large numbers",
      "arbitrary transient random walk",
      "spatial sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}