{
  "id": "1009.3406",
  "version": "v1",
  "published": "2010-09-17T12:09:09.000Z",
  "updated": "2010-09-17T12:09:09.000Z",
  "title": "The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation",
  "authors": [
    "Emanuele Dolera",
    "Eugenio Regazzini"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/09-AAP623 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2010, Vol. 20, No. 2, 430-461",
  "doi": "10.1214/09-AAP623",
  "categories": [
    "math.PR"
  ],
  "abstract": "In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186-201] it is proved that the total variation distance between the solution $f(\\cdot,t)$ of Kac's equation and the Gaussian density $(0,\\sigma^2)$ has an upper bound which goes to zero with an exponential rate equal to -1/4 as $t\\to+\\infty$. In the present paper, we determine a lower bound which decreases exponentially to zero with this same rate, provided that a suitable symmetrized form of $f_0$ has nonzero fourth cumulant $\\kappa_4$. Moreover, we show that upper bounds like $\\bar{C}_{\\delta}e^{-({1/4})t}\\rho_{\\delta}(t)$ are valid for some $\\rho_{\\delta}$ vanishing at infinity when $\\int_{\\mathbb{R}}|v|^{4+\\delta}f_0(v)\\,dv<+\\infty$ for some $\\delta$ in $[0,2[$ and $\\kappa_4=0$. Generalizations of this statement are presented, together with some remarks about non-Gaussian initial conditions which yield the insuperable barrier of -1 for the rate of convergence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-17T12:09:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central limit theorem",
      "discovering sharp rates",
      "kac equation",
      "convergence",
      "equilibrium"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.3406D"
    }
  }
}