{
  "id": "math/0406015",
  "version": "v2",
  "published": "2004-06-01T13:55:54.000Z",
  "updated": "2006-06-30T07:08:06.000Z",
  "title": "Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes",
  "authors": [
    "M. E. Caballero",
    "L. Chaumont"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117905000000611 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 3, 1012-1034",
  "doi": "10.1214/009117905000000611",
  "categories": [
    "math.PR"
  ],
  "abstract": "Using Lamperti's relationship between L\\'{e}vy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law $\\mathbb{P}_x$ of a pssMp starting at $x>0$, in the Skorohod space of c\\`{a}dl\\`{a}g paths, when $x$ tends to 0. To do so, we first give conditions which allow us to construct a c\\`{a}dl\\`{a}g Markov process $X^{(0)}$, starting from 0, which stays positive and verifies the scaling property. Then we establish necessary and sufficient conditions for the laws $\\mathbb{P}_x$ to converge weakly to the law of $X^{(0)}$ as $x$ goes to 0. In particular, this answers a question raised by Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205--225] about the Feller property for pssMp at $x=0$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-06-30T07:08:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G18",
      "60G51",
      "60B10"
    ],
    "keywords": [
      "positive self-similar markov processes",
      "weak convergence",
      "lévy processes",
      "overshoots",
      "feller property"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6015C"
    }
  }
}