{
  "id": "1012.0093",
  "version": "v1",
  "published": "2010-12-01T04:45:31.000Z",
  "updated": "2010-12-01T04:45:31.000Z",
  "title": "Description of limits of ranges of iterations of stochastic integral mappings of infinitely divisible distributions",
  "authors": [
    "Ken-iti Sato"
  ],
  "comment": "16 pages. To appear in ALEA Lat. Am. J. Probab. Math. Statist",
  "categories": [
    "math.PR"
  ],
  "abstract": "For infinitely divisible distributions $\\rho$ on $\\mathbb{R}^d$ the stochastic integral mapping $\\Phi_f\\rho$ is defined as the distribution of improper stochastic integral $\\int_0^{\\infty-} f(s) dX_s^{(\\rho)}$, where $f(s)$ is a non-random function and $\\{X_s^{(\\rho)}\\}$ is a L\\'evy process on $\\mathbb{R}^d$ with distribution $\\rho$ at time 1. For three families of functions $f$ with parameters, the limits of the nested sequences of the ranges of the iterations $\\Phi_f^n$ are shown to be some subclasses, with explicit description, of the class $L_{\\infty}$ of completely selfdecomposable distributions. In the critical case of parameter 1, the notion of weak mean 0 plays an important role. Examples of $f$ with different limits of the ranges of $\\Phi_f^n$ are also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-01T04:45:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E07",
      "60G51",
      "60H05"
    ],
    "keywords": [
      "infinitely divisible distributions",
      "stochastic integral mapping",
      "iterations",
      "improper stochastic integral",
      "levy process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.0093S"
    }
  }
}