{
  "id": "1711.05384",
  "version": "v1",
  "published": "2017-11-15T02:27:44.000Z",
  "updated": "2017-11-15T02:27:44.000Z",
  "title": "Normal Approximation by Stein's Method under Sublinear Expectations",
  "authors": [
    "Yongsheng Song"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Peng (2008)(\\cite{P08b}) proved the Central Limit Theorem under a sublinear expectation: \\textit{Let $(X_i)_{i\\ge 1}$ be a sequence of i.i.d random variables under a sublinear expectation $\\hat{\\mathbf{E}}$ with $\\hat{\\mathbf{E}}[X_1]=\\hat{\\mathbf{E}}[-X_1]=0$ and $\\hat{\\mathbf{E}}[|X_1|^3]<\\infty$. Setting $W_n:=\\frac{X_1+\\cdots+X_n}{\\sqrt{n}}$, we have, for each bounded and Lipschitz function $\\varphi$, \\[\\lim_{n\\rightarrow\\infty}\\bigg|\\hat{\\mathbf{E}}[\\varphi(W_n)]-\\mathcal{N}_G(\\varphi)\\bigg|=0,\\] where $\\mathcal{N}_G$ is the $G$-normal distribution with $G(a)=\\frac{1}{2}\\hat{\\mathbf{E}}[aX_1^2]$, $a\\in \\mathbb{R}$.} In this paper, we shall give an estimate of the rate of convergence of this CLT by Stein's method under sublinear expectations: \\textit{Under the same conditions as above, there exists $\\alpha\\in(0,1)$ depending on $\\underline{\\sigma}$ and $\\overline{\\sigma}$, and a positive constant $C_{\\alpha, G}$ depending on $\\alpha, \\underline{\\sigma}$ and $\\overline{\\sigma}$ such that \\[\\sup_{|\\varphi|_{Lip}\\le1}\\bigg|\\hat{\\mathbf{E}}[\\varphi(W_n)]-\\mathcal{N}_G(\\varphi)\\bigg|\\leq C_{\\alpha,G}\\frac{\\hat{\\mathbf{E}}[|X_1|^{2+\\alpha}]}{n^{\\frac{\\alpha}{2}}},\\] where $\\overline{\\sigma}^2=\\hat{\\mathbf{E}}[X_1^2]$, $\\underline{\\sigma}^2=-\\hat{\\mathbf{E}}[-X_1^2]>0$ and $\\mathcal{N}_G$ is the $G$-normal distribution with \\[G(a)=\\frac{1}{2}\\hat{\\mathbf{E}}[aX_1^2]=\\frac{1}{2}(\\overline{\\sigma}^2a^+-\\underline{\\sigma}^2a^-), \\ a\\in \\mathbb{R}.\\]}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-15T02:27:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60G50"
    ],
    "keywords": [
      "sublinear expectation",
      "steins method",
      "normal approximation",
      "normal distribution",
      "central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}