{
  "id": "1703.07868",
  "version": "v1",
  "published": "2017-03-22T21:56:26.000Z",
  "updated": "2017-03-22T21:56:26.000Z",
  "title": "A probability inequality for sums of independent Banach space valued random variables",
  "authors": [
    "Deli Li",
    "Han-Ying Liang",
    "Andrew Rosalsky"
  ],
  "comment": "10 pages. arXiv admin note: substantial text overlap with arXiv:1506.07596",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(\\mathbf{B}, \\|\\cdot\\|)$ be a real separable Banach space. Let $\\varphi(\\cdot)$ and $\\psi(\\cdot)$ be two continuous and increasing functions defined on $[0, \\infty)$ such that $\\varphi(0) = \\psi(0) = 0$, $\\lim_{t \\rightarrow \\infty} \\varphi(t) = \\infty$, and $\\frac{\\psi(\\cdot)}{\\varphi(\\cdot)}$ is a nondecreasing function on $[0, \\infty)$. Let $\\{V_{n};~n \\geq 1 \\}$ be a sequence of independent and symmetric {\\bf B}-valued random variables. In this note, we establish a probability inequality for sums of independent {\\bf B}-valued random variables by showing that for every $n \\geq 1$ and all $t \\geq 0$, \\[ \\mathbb{P}\\left(\\left\\|\\sum_{i=1}^{n} V_{i} \\right\\| > t b_{n} \\right) \\leq 4 \\mathbb{P} \\left(\\left\\|\\sum_{i=1}^{n} \\varphi\\left(\\psi^{-1}(\\|V_{i}\\|)\\right) \\frac{V_{i}}{\\|V_{i}\\|} \\right\\| > t a_{n} \\right) + \\sum_{i=1}^{n}\\mathbb{P}\\left(\\|V_{i}\\| > b_{n} \\right), \\] where $a_{n} = \\varphi(n)$ and $b_{n} = \\psi(n)$, $n \\geq 1$. As an application of this inequality, we establish what we call a comparison theorem for the weak law of large numbers for independent and identically distributed ${\\bf B}$-valued random variables.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-22T21:56:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "60B12",
      "60G50"
    ],
    "keywords": [
      "banach space valued random variables",
      "independent banach space valued random",
      "probability inequality",
      "real separable banach space",
      "comparison theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}