{
  "id": "1205.1284",
  "version": "v1",
  "published": "2012-05-07T05:13:40.000Z",
  "updated": "2012-05-07T05:13:40.000Z",
  "title": "An Inequality Related to Negative Definite Functions",
  "authors": [
    "M. Lifshits",
    "R. L. Schilling",
    "I. Tyurin"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "This is a substantially generalized version of the preprint arXiv:1105.4214 by Lifshits and Tyurin. We prove that for any pair of i.i.d. random vectors $X, Y$ in $R^n$ and any real-valued continuous negative definite function $g: R^n\\to R$ the inequality $$ E g(X-Y) \\le E g(X+Y)$$ holds. In particular, for $a \\in (0,2]$ and the Euclidean norm $|.|$ one has $$ E |X-Y|^a \\le E |X+Y|^a. $$ The latter inequality is due to A. Buja et al. (Ann. Statist., 1994} where it is used for some applications in multivariate statistics. We show a surprising connection with bifractional Brownian motion and provide some related counter-examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-07T05:13:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "60G22",
      "60E10"
    ],
    "keywords": [
      "inequality",
      "bifractional brownian motion",
      "euclidean norm",
      "random vectors",
      "real-valued continuous negative definite function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.1284L"
    }
  }
}