{
  "id": "1301.5713",
  "version": "v2",
  "published": "2013-01-24T07:01:21.000Z",
  "updated": "2013-06-28T09:05:32.000Z",
  "title": "Some aspects of fluctuations of random walks on R and applications to random walks on R+ with non-elastic reflection at 0",
  "authors": [
    "Rim Essifi",
    "Marc Peigné",
    "Kilian Raschel"
  ],
  "comment": "17 pages, 1 figure",
  "journal": "ALEA, Latin American Journal of Probability and Mathematical Statistics 10 (2) (2013) 591-607",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article we refine well-known results concerning the fluctuations of one-dimensional random walks. More precisely, if $(S_n)_{n \\geq 0}$ is a random walk starting from 0 and $r\\geq 0$, we obtain the precise asymptotic behavior as $n\\to\\infty$ of $\\mathbb P[\\tau^{>r}=n, S_n\\in K]$ and $\\mathbb P[\\tau^{>r}>n, S_n\\in K]$, where $\\tau^{>r}$ is the first time that the random walk reaches the set $]r,\\infty[$, and $K$ is a compact set. Our assumptions on the jumps of the random walks are optimal. Our results give an answer to a question of Lalley stated in [9], and are applied to obtain the asymptotic behavior of the return probabilities for random walks on $\\mathbb R^+$ with non-elastic reflection at 0.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-28T09:05:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60G50",
      "31C05"
    ],
    "keywords": [
      "non-elastic reflection",
      "fluctuations",
      "applications",
      "one-dimensional random walks",
      "precise asymptotic behavior"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.5713E"
    }
  }
}