{
  "id": "1610.04471",
  "version": "v1",
  "published": "2016-10-14T14:09:01.000Z",
  "updated": "2016-10-14T14:09:01.000Z",
  "title": "Zooming in on a Lévy process at its supremum",
  "authors": [
    "Jevgenijs Ivanovs"
  ],
  "comment": "15 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $M$ and $\\tau$ be the supremum and its time of a L\\'evy process $X$ on some finite time interval. It is shown that zooming in on $X$ at its supremum, that is, considering $(a_\\eta(X_{\\tau+t/\\eta}-M))_{t\\in\\mathbb R}$ as $\\eta,a_\\eta\\rightarrow\\infty$, results in $(\\xi_t)_{t\\in\\mathbb R}$ constructed from two independent processes corresponding to some self-similar L\\'evy process $S$ conditioned to stay positive and negative. This holds when $X$ is in the domain of attraction of $S$ under the zooming-in procedure as opposed to the classical zooming-out of Lamperti (1962). As an application of this result we provide a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result of Asmussen, Glynn and Pitman (1995) for the Brownian motion. Moreover, a general invariance principle for L\\'evy processes conditioned to stay negative is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-14T14:09:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60F17",
      "60G18",
      "60G52"
    ],
    "keywords": [
      "lévy process",
      "finite time interval",
      "self-similar levy process",
      "general invariance principle",
      "zooming-in procedure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}