{
  "id": "1902.04196",
  "version": "v1",
  "published": "2019-02-12T00:42:54.000Z",
  "updated": "2019-02-12T00:42:54.000Z",
  "title": "The Poincaré inequality and quadratic transportation-variance inequalities",
  "authors": [
    "Yuan Liu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "It is known that the Poincar\\'e inequality is equivalent to the quadratic transportation-variance inequality (namely $W_2^2(f\\mu,\\mu) \\leqslant C_V \\mathrm{Var}_\\mu(f)$), see Ledoux \\cite{Ledoux18} most recently. We give an alternative proof to this fact. In particular, we achieve a smaller $C_V$ than before, which exactly coincides with the double of Poincar\\'e constant. Applying the same argument leads to more characterizations of the Poincar\\'e inequality. Next, we present a comparison inequality between $W_2^2(f\\mu,\\mu)$ and its centralization $W_2^2(f_c\\mu,\\mu)$ for $f_c = \\frac{|\\sqrt{f} - \\mu(\\sqrt{f})|^2}{\\mathrm{Var}_\\mu (\\sqrt{f})}$, which may be viewed as some special counterpart of the Rothaus' lemma for relative entropy. Then it yields some new bound of $W_2^2(f\\mu,\\mu)$ associated to the variance of $\\sqrt{f}$ rather than $f$. As a byproduct, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, with no requirement of the Bobkov-G\\\"otze's characterization of the Talagrand's inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-12T00:42:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D10",
      "60E15",
      "60J60"
    ],
    "keywords": [
      "quadratic transportation-variance inequality",
      "poincare inequality",
      "quadratic transportation-information inequality",
      "lyapunov condition",
      "talagrands inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}