{
  "id": "0712.0728",
  "version": "v1",
  "published": "2007-12-05T14:13:52.000Z",
  "updated": "2007-12-05T14:13:52.000Z",
  "title": "Asymptotics for first-passage times of Lévy processes and random walks",
  "authors": [
    "Denis Denisov",
    "Vsevolod Shneer"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the exact asymptotics for the distribution of the first time $\\tau_x$ a L\\'evy process $X_t$ crosses a negative level $-x$. We prove that $\\mathbf P(\\tau_x>t)\\sim V(x)\\mathbf P(X_t\\ge 0)/t$ as $t\\to\\infty$ for a certain function $V(x)$. Using known results for the large deviations of random walks we obtain asymptotics for $\\mathbf P(\\tau_x>t)$ explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-05T14:13:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60G51"
    ],
    "keywords": [
      "random walks",
      "lévy processes",
      "first-passage times",
      "first time",
      "exact asymptotics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.0728D"
    }
  }
}