{
  "id": "2408.10662",
  "version": "v1",
  "published": "2024-08-20T09:03:21.000Z",
  "updated": "2024-08-20T09:03:21.000Z",
  "title": "Convergence rate in the law of logarithm for negatively dependent random variables under sub-linear expectations",
  "authors": [
    "Mingzhou Xu",
    "Wei Wang"
  ],
  "comment": "8 pages, submitted to Mathematica Applicata",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{X,X_n,n\\ge 1\\}$ be a sequence of identically distributed, negatively dependent (NA) random variables under sub-linear expectations, and denote $S_n=\\sum_{i=1}^{n}X_i$, $n\\ge 1$. Assume that $h(\\cdot)$ is a positive non-decreasing function on $(0,\\infty)$ fulfulling $\\int_{1}^{\\infty}(th(t))^{-1}\\dif t=\\infty$. Write $Lt=\\ln \\max\\{\\me,t\\}$, $\\psi(t)=\\int_{1}^{t}(sh(s))^{-1}\\dif s$, $t\\ge 1$. In this sequel, we establish that $\\sum_{n=1}^{\\infty}(nh(n))^{-1}\\vv\\left\\{|S_n|\\ge (1+\\varepsilon)\\sigma\\sqrt{2nL\\psi(n)}\\right\\}<\\infty$, $\\forall \\varepsilon>0$ if $\\ee(X)=\\ee(-X)=0$ and $\\ee(X^2)=\\sigma^2\\in (0,\\infty)$. The result generalizes that of NA random variables in probability space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-20T09:03:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60G50"
    ],
    "keywords": [
      "negatively dependent random variables",
      "sub-linear expectations",
      "convergence rate",
      "na random variables",
      "result generalizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}