{
  "id": "1707.00829",
  "version": "v1",
  "published": "2017-07-04T06:58:22.000Z",
  "updated": "2017-07-04T06:58:22.000Z",
  "title": "A functional limit theorem for random processes with immigration in the case of heavy tails",
  "authors": [
    "Alexander Marynych",
    "Glib Verovkin"
  ],
  "comment": "Published at http://dx.doi.org/10.15559/17-VMSTA76 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)",
  "journal": "Modern Stochastics: Theory and Applications 2017, Vol. 4, No. 2, 93-108",
  "doi": "10.15559/17-VMSTA76",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(X_k,\\xi_k)_{k\\in \\mathbb {N}}$ be a sequence of independent copies of a pair $(X,\\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\\infty)$ and $\\xi$ is a positive random variable. The random process with immigration $(Y(u))_{u\\in \\mathbb {R}}$ is defined as the a.s. finite sum $Y(u)=\\sum_{k\\geq0}X_{k+1}(u- \\xi_1-\\cdots-\\xi_k)1\\mkern-4.5mu\\mathrm{l}_{\\{\\xi_1+\\cdots+\\xi_k\\leq u\\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\\geq 0}$, as $t\\to\\infty$, when the law of $\\xi$ belongs to the domain of attraction of an $\\alpha$-stable law with $\\alpha\\in(0,1)$, and the process $X$ oscillates moderately around its mean $\\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\\geq0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\\infty)$ to a fractionally integrated inverse stable subordinator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-04T06:58:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "functional limit theorem",
      "random process",
      "heavy tails",
      "integrated inverse stable subordinator",
      "immigration"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}