{
  "id": "math/0211409",
  "version": "v1",
  "published": "2002-11-26T13:57:18.000Z",
  "updated": "2002-11-26T13:57:18.000Z",
  "title": "Cramer's estimate for the exponential functional of a Levy process",
  "authors": [
    "Mejane Olivier"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the exponential functional $A_{\\infty}=\\int_0^{\\infty} e^{\\xi_s} ds$ associated to a Levy process $(\\xi_t)_{t \\geq 0}$. We find the asymptotic behavior of the tail of this random variable, under some assumptions on the process $\\xi$, the main one being Cramer's condition, that asserts the existence of a real $\\chi >0$ such that ${\\Bbb E}(e^{\\chi \\xi_1})=1$. Then there exists $C>0$ satisfying, when $t \\to +\\infty$ : $$ {\\Bbb P} (A_{\\infty}> t) \\sim C t^{-\\chi} \\quad . $$ This result can be applied for example to the process $\\xi_t = at - S_{\\alpha}(t)$ where $S_{\\alpha}$ stands for the stable subordinator of index $\\alpha$ ($0 < \\alpha < 1$), and $a$ is a positive real (we have then $\\chi=a^{1/(\\alpha -1)}$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-11-26T13:57:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "levy process",
      "exponential functional",
      "cramers estimate",
      "asymptotic behavior",
      "cramers condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11409O"
    }
  }
}