{
  "id": "1702.05892",
  "version": "v1",
  "published": "2017-02-20T08:23:32.000Z",
  "updated": "2017-02-20T08:23:32.000Z",
  "title": "On the rate of convergence in the central limit theorem for hierarchical Laplacian",
  "authors": [
    "Alexander Bendikov",
    "Wojciech Cygan"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ we consider the hierarchical Laplacian $L=L_{C}$. Choosing a sequence $\\{\\varepsilon (B)\\}$ of i.i.d. random variables we define the perturbed function $C(B,\\omega )$ and the perturbed hierarchical Laplacian $L^{\\omega }=L_{C(\\omega )}.$ We study the arithmetic means $\\overline{\\lambda }(\\omega )$ of the $L^{\\omega }$-eigenvalues. Under some mild assumptions the normalized arithmetic means $\\big( \\overline{\\lambda }-\\mathbb{E}\\overline{\\lambda }\\big) /\\sigma \\big( \\overline{\\lambda }\\big) $ converge in law to the standard normal distribution. In this note we study convergence in the total variation distance and estimate the rate of convergence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-20T08:23:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12H25",
      "60F05",
      "94A17",
      "47S10",
      "60J25"
    ],
    "keywords": [
      "central limit theorem",
      "hierarchical laplacian",
      "convergence",
      "proper ultrametric space",
      "standard normal distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}