{
  "id": "1509.06149",
  "version": "v1",
  "published": "2015-09-21T08:48:36.000Z",
  "updated": "2015-09-21T08:48:36.000Z",
  "title": "Self-normalized moderate deviation and laws of the iterated logarithm under G-expectation",
  "authors": [
    "Li-Xin Zhang"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1507.07600",
  "categories": [
    "math.PR"
  ],
  "abstract": "The sub-linear expectation or called G-expectation is a nonlinear expectation having advantage of modeling non-additive probability problems and the volatility uncertainty in finance. Let $\\{X_n;n\\ge 1\\}$ be a sequence of independent random variables in a sub-linear expectation space $(\\Omega, \\mathscr{H}, \\widehat{\\mathbb E})$. Denote $S_n=\\sum_{k=1}^n X_k$ and $V_n^2=\\sum_{k=1}^n X_k^2$. In this paper, a moderate deviation for self-normalized sums, that is, the asymptotic capacity of the event $\\{S_n/V_n \\ge x_n \\}$ for $x_n=o(\\sqrt{n})$, is found both for identically distributed random variables and independent but not necessarily identically distributed random variables. As an applications, the self-normalized laws of the iterated logarithm are obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-21T08:48:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60F05",
      "60H10",
      "60G48"
    ],
    "keywords": [
      "self-normalized moderate deviation",
      "iterated logarithm",
      "g-expectation",
      "sub-linear expectation space",
      "independent random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150906149Z"
    }
  }
}