{
  "id": "1609.01435",
  "version": "v1",
  "published": "2016-09-06T08:44:18.000Z",
  "updated": "2016-09-06T08:44:18.000Z",
  "title": "Operator self-similar processes and functional central limit theorems",
  "authors": [
    "Vaidotas Characiejus",
    "Alfredas Račkauskas"
  ],
  "comment": "22 pages",
  "journal": "Stochastic Processes and their Applications, Volume 124, Issue 8, August 2014, Pages 2605-2627",
  "doi": "10.1016/j.spa.2014.03.007",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{X_k:k\\ge1\\}$ be a linear process with values in the separable Hilbert space $L_2(\\mu)$ given by $X_k=\\sum_{j=0}^\\infty(j+1)^{-D}\\varepsilon_{k-j}$ for each $k\\ge1$, where $D$ is defined by $Df=\\{d(s)f(s):s\\in\\mathbb S\\}$ for each $f\\in L_2(\\mu)$ with $d:\\mathbb S\\to\\mathbb R$ and $\\{\\varepsilon_k:k\\in\\mathbb Z\\}$ are independent and identically distributed $L_2(\\mu)$-valued random elements with $\\operatorname E\\varepsilon_0=0$ and $\\operatorname E\\|\\varepsilon_0\\|^2<\\infty$. We establish sufficient conditions for the functional central limit theorem for $\\{X_k:k\\ge1\\}$ when the series of operator norms $\\sum_{j=0}^\\infty\\|(j+1)^{-D}\\|$ diverges and show that the limit process generates an operator self-similar process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-06T08:44:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B12",
      "60F17",
      "60G18"
    ],
    "keywords": [
      "functional central limit theorem",
      "operator self-similar process",
      "limit process generates",
      "establish sufficient conditions",
      "separable hilbert space"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}