{
  "id": "1305.6245",
  "version": "v3",
  "published": "2013-05-27T14:51:12.000Z",
  "updated": "2014-03-10T13:05:07.000Z",
  "title": "Lévy processes with marked jumps I : Limit theorems",
  "authors": [
    "Cécile Delaporte"
  ],
  "comment": "27 pages. Final version accepted for publication in Journal of Theoretical Probability",
  "doi": "10.1007/s10959-014-0549-9",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a sequence (Z_n,Z_n^M) of bivariate L\\'evy processes, such that Z_n is a spectrally positive L\\'evy process with finite variation, and Z_n^M is the counting process of marks in {0,1} carried by the jumps of Z_n. The study of these processes is justified by their interpretation as contour processes of a sequence of splitting trees with mutations at birth. Indeed, this paper is the first part of a work aiming to establish an invariance principle for the genealogies of such populations enriched with their mutational histories. To this aim, we define a bivariate subordinator that we call the marked ladder height process of (Z_n,Z_n^M), as a generalization of the classical ladder height process to our L\\'evy processes with marked jumps. Assuming that the sequence (Z_n) converges towards a L\\'evy process Z with infinite variation, we first prove the convergence in distribution, with two possible regimes for the marks, of the marked ladder height process of (Z_n,Z_n^M). Then we prove the joint convergence in law of Z_n with its local time at the supremum and its marked ladder height process.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-03-10T13:05:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60J55",
      "60G51"
    ],
    "keywords": [
      "marked ladder height process",
      "marked jumps",
      "limit theorems",
      "lévy processes",
      "bivariate levy processes"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6245D"
    }
  }
}