{
  "id": "2310.16657",
  "version": "v1",
  "published": "2023-10-25T14:13:45.000Z",
  "updated": "2023-10-25T14:13:45.000Z",
  "title": "The asymptotic behavior of rarely visited edges of the simple random walk",
  "authors": [
    "Ze-Chun Hu",
    "Xue Peng",
    "Renming Song",
    "Yuan Tan"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\\mathbb{Z}$. Let $\\alpha(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\\mathbb{E}(\\alpha(n))$, show that $n\\to \\mathbb{E}(\\alpha(n))$ is non-decreasing in $n$ and that $\\lim\\limits_{n\\to+\\infty}\\mathbb{E}(\\alpha(n))=2$. Then we study the asymptotic behavior of $\\mathbb{P} (\\alpha(n)>a(\\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\\in(0,+\\infty)$ such that $\\limsup\\limits_{n\\to+\\infty}\\frac{\\alpha(n)}{(\\log n)^2}=C$ almost surely.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-25T14:13:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rarely visited edges",
      "asymptotic behavior",
      "simple random walk",
      "simple symmetric random walk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}