{
  "id": "0906.3886",
  "version": "v2",
  "published": "2009-06-21T18:06:25.000Z",
  "updated": "2011-06-17T05:18:33.000Z",
  "title": "Concentration of measures via size biased couplings",
  "authors": [
    "Subhankar Ghosh",
    "Larry Goldstein"
  ],
  "comment": "Concentration results for the number of isolated vertices have been removed from this version, and with corrections, posted jointly with Martin Raic in http://arxiv.org/abs/1106.0048",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $Y$ be a nonnegative random variable with mean $\\mu$ and finite positive variance $\\sigma^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by E[Yf(Y)]=\\mu E f(Y^s) for all functions $f$ for which these expectations exist. Under a variety of conditions on the coupling of Y and $Y^s$, including combinations of boundedness and monotonicity, concentration of measure inequalities hold. Examples include the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of m-runs in a sequence of coin tosses, the number of local maximum of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, the volume covered by the union of $n$ balls placed uniformly over a volume n subset of d dimensional Euclidean space, the number of bulbs switched on at the terminal time in the so called lightbulb process, and the infinitely divisible and compound Poisson distributions that satisfy a bounded moment generating function condition.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-17T05:18:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "60C05"
    ],
    "keywords": [
      "size biased couplings",
      "concentration",
      "compound poisson distributions",
      "dimensional euclidean space",
      "urn allocation model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.3886G"
    }
  }
}