{
  "id": "math/0603030",
  "version": "v3",
  "published": "2006-03-01T19:58:14.000Z",
  "updated": "2006-03-13T19:01:09.000Z",
  "title": "On inequalities for sums of bounded random variables",
  "authors": [
    "Iosif Pinelis"
  ],
  "comment": "6 pages; the result in the previous version is strengthened and extended",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let $\\eta_{1},\\eta_2,...$ be independent (not necessarily identically distributed) zero-mean random variables (r.v.'s) such that $|\\eta_i|\\le1$ almost surely for all $i$, and let $Z$ stand for a standard normal r.v. Let $a_1,a_2,...$ be any real numbers such that $a_1^2+a_2^2+...=1.$ It is shown that then $$ \\P(a_1\\eta_1+a_2\\eta_2+...\\ge x) \\le \\P(Z\\ge x-\\la/x) \\forall x>0, $$ where $\\la := \\ln\\frac{2e^3}9=1.495...$. The proof relies on (i) another probability inequality and (ii) a l'Hospital-type rule for monotonicity, both developed elsewhere. A multidimensional analogue of this result is given, based on a dimensionality reduction device, also developed elsewhere. In addition, extensions to (super)martingales are indicated.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-03-13T19:01:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "60G50",
      "60G42",
      "60G48",
      "26A48",
      "26D10"
    ],
    "keywords": [
      "bounded random variables",
      "zero-mean random variables",
      "dimensionality reduction device",
      "proof relies",
      "probability inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3030P"
    }
  }
}