{
  "id": "math/0601652",
  "version": "v1",
  "published": "2006-01-26T18:45:16.000Z",
  "updated": "2006-01-26T18:45:16.000Z",
  "title": "Symmetrization of Bernoulli",
  "authors": [
    "Soumik Pal"
  ],
  "comment": "3 pages; a completely probabilistic proof of a theorem due to Kagan, Mallows, Shepp, Vanderbei & Vardi",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let X be a random variable. We shall call an independent random variable Y to be a symmetrizer for X, if X+Y is symmetric around zero. A random variable is said to be symmetry resistant if the variance of any symmetrizer Y, is never smaller than the variance of X itself. We prove that a Bernoulli(p) random variable is symmetry resistant if and only if p is not 1/2. This is an old problem proved in 1999 by Kagan, Mallows, Shepp, Vanderbei & Vardi using linear programming principles. We reprove it here using completely probabilistic tools using Skorokhod embedding and Ito's rule.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-26T18:45:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G99"
    ],
    "keywords": [
      "random variable",
      "symmetry resistant",
      "symmetrization",
      "symmetrizer",
      "independent random"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1652P"
    }
  }
}