{
  "id": "2207.04611",
  "version": "v1",
  "published": "2022-07-11T04:06:43.000Z",
  "updated": "2022-07-11T04:06:43.000Z",
  "title": "A Strong Law of Large Numbers under Sublinear Expectations",
  "authors": [
    "Yongsheng Song"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a sequence of i.i.d. random variables $\\{\\xi_k\\}$under a sublinear expectation $\\mathbb{E}=\\sup_{P\\in\\Theta}E_P$. We first give a new proof to the fact that, under each $P\\in\\Theta$, any cluster point of the empirical averages $\\bar{\\xi}_n=(\\xi_1+\\cdots+\\xi_n)/n$ lies in $[\\underline{\\mu}, \\bar{\\mu}]$ with $\\underline{\\mu}=-\\mathbb{E}[-\\xi_1], \\bar{\\mu}=\\mathbb{E}[\\xi_1]$. Then, we consider sublinear expectations on a Polish space $\\Omega$, and show that for each constant $\\mu\\in [\\underline{\\mu},\\bar{\\mu}]$, there exists a probability $P_{\\mu}\\in\\Theta$ such that \\begin {eqnarray}\\label {intro-a.s.} \\lim_{n\\rightarrow\\infty}\\bar{\\xi}_n=\\mu, \\ P_{\\mu}\\textmd{-a.s.}, \\end {eqnarray} supposing that $\\Theta$ is weakly compact and $\\{\\xi_n\\}\\in L^1_{\\mathbb{E}}(\\Omega)$. Under the same conditions, we can get a generalization of (\\ref {intro-a.s.}) in the product space $\\Omega=\\mathbb{R}^{\\mathbb{N}}$ with $\\mu\\in [\\underline{\\mu},\\bar{\\mu}]$ replaced by $\\Pi=\\pi(\\xi_1, \\cdots,\\xi_d)\\in [\\underline{\\mu},\\bar{\\mu}]$, where $\\pi$ is a Borel measurable function on $\\mathbb{R}^d$, $d\\in\\mathbb{R}$. Finally, we characterize the triviality of the tail $\\sigma$-algebra of i.i.d. random variables under a sublinear expectation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-11T04:06:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60G50",
      "60G65"
    ],
    "keywords": [
      "sublinear expectation",
      "large numbers",
      "strong law",
      "random variables",
      "cluster point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}