{
  "id": "1004.1850",
  "version": "v8",
  "published": "2010-04-11T22:27:34.000Z",
  "updated": "2011-06-28T08:32:42.000Z",
  "title": "Level-crossings of symmetric random walks and their application",
  "authors": [
    "Vyacheslav M. Abramov"
  ],
  "comment": "Substantially revised, will be submitted",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_1$, $X_2$, $...$ be a sequence of independently and identically distributed random variables with $\\mathsf{E}X_1=0$, and let $S_0=0$ and $S_t=S_{t-1}+X_t$, $t=1,2,...$, be a random walk. Denote $\\tau={cases}\\inf\\{t>1: S_t\\leq0\\}, &\\text{if} \\ X_1>0, 1, &\\text{otherwise}. {cases}$ Let $\\alpha$ denote a positive number, and let $L_\\alpha$ denote the number of level-crossings from the below (or above) across the level $\\alpha$ during the interval $[0, \\tau]$. Under quite general assumption, an inequality for the expected number of level-crossings is established. Under some special assumptions, it is proved that there exists an infinitely increasing sequence $\\alpha_n$ such that the equality $\\mathsf{E}L_{\\alpha_n}=c\\mathsf{P}\\{X_1>0\\}$ is satisfied, where $c$ is a specified constant that does not depend on $n$. The result is illustrated for a number of special random walks. We also give non-trivial examples from queuing theory where the results of this theory are applied.",
  "revisions": [
    {
      "version": "v8",
      "updated": "2011-06-28T08:32:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60K25"
    ],
    "keywords": [
      "symmetric random walks",
      "level-crossings",
      "application",
      "quite general assumption",
      "special random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.1850A"
    }
  }
}