{
  "id": "1501.07486",
  "version": "v1",
  "published": "2015-01-29T15:42:52.000Z",
  "updated": "2015-01-29T15:42:52.000Z",
  "title": "Weak dependence for a class of local functionals of Markov chains on $\\mathbb{Z}^d$",
  "authors": [
    "Carlo Boldrighini",
    "Antonella Marchesiello",
    "Chiara Saffirio"
  ],
  "comment": "16 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "In some papers on infinite Markov chains in $\\mathbb{Z}^d$, and notably in the work of R.A. Minlos and collaborators, one can prove the existence of a spectral gap for a suitable subspace of local functions. We consider functions of the type $f(\\widehat \\eta)$, where $\\widehat \\eta= \\{\\eta_{t}\\}_{t=0}^{\\infty}$ is the sequence of the states, and $f$ is local. In the case of a simple example of random walk in random environment with mutual interaction we show that there is a natural class of functions $f$, related to the H\\\"older continuos functions $\\mathcal C^{\\alpha}$on the torus $T^{1}$, with $\\alpha\\in (0,1)$ large enough, depending on the spectral gap, for which the Central Limit Theorem holds for the sequence $f(S^{k}\\widehat \\eta)$, $k=0,1,\\ldots$, where $S$ is the time shift.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-29T15:42:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local functionals",
      "weak dependence",
      "spectral gap",
      "central limit theorem holds",
      "infinite markov chains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150107486B"
    }
  }
}