{
  "id": "math/0410168",
  "version": "v1",
  "published": "2004-10-06T15:37:52.000Z",
  "updated": "2004-10-06T15:37:52.000Z",
  "title": "Measure concentration for Euclidean distance in the case of dependent random variables",
  "authors": [
    "Katalin Marton"
  ],
  "comment": "Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000702",
  "journal": "Annals of Probability 2004, Vol. 32, No. 3B, 2526-2544",
  "doi": "10.1214/009117904000000702",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let q^n be a continuous density function in n-dimensional Euclidean space. We think of q^n as the density function of some random sequence X^n with values in \\BbbR^n. For I\\subset[1,n], let X_I denote the collection of coordinates X_i, i\\in I, and let \\bar X_I denote the collection of coordinates X_i, i\\notin I. We denote by Q_I(x_I|\\bar x_I) the joint conditional density function of X_I, given \\bar X_I. We prove measure concentration for q^n in the case when, for an appropriate class of sets I, (i) the conditional densities Q_I(x_I|\\bar x_I), as functions of x_I, uniformly satisfy a logarithmic Sobolev inequality and (ii) these conditional densities also satisfy a contractivity condition related to Dobrushin and Shlosman's strong mixing condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-06T15:37:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82C22"
    ],
    "keywords": [
      "dependent random variables",
      "measure concentration",
      "euclidean distance",
      "joint conditional density function",
      "shlosmans strong mixing condition"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10168M"
    }
  }
}