{
  "id": "0903.3705",
  "version": "v1",
  "published": "2009-03-22T05:30:08.000Z",
  "updated": "2009-03-22T05:30:08.000Z",
  "title": "Invariance principles for local times at the supremum of random walks and Lévy processes",
  "authors": [
    "Loïc Chaumont",
    "Ron Arthur Doney"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove that when a sequence of L\\'evy processes $X^{(n)}$ or a normed sequence of random walks $S^{(n)}$ converges a.s. on the Skorokhod space toward a L\\'evy process $X$, the sequence $L^{(n)}$ of local times at the supremum of $X^{(n)}$ converges uniformly on compact sets in probability toward the local time at the supremum of $X$. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law towards the ladder processes of $X$. As an application, we show that in general, the sequence $S^{(n)}$ conditioned to stay positive converges weakly, jointly with its local time at the future minimum, towards the corresponding functional for the limiting process $X$. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-22T05:30:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local time",
      "random walks",
      "invariance principle",
      "lévy processes",
      "ladder processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.3705C"
    }
  }
}