{
  "id": "1205.4866",
  "version": "v1",
  "published": "2012-05-22T10:12:42.000Z",
  "updated": "2012-05-22T10:12:42.000Z",
  "title": "Uniform oscillatory behavior of spherical functions of $GL_n/U_n$ at the identity and a central limit theorem",
  "authors": [
    "Michael Voit"
  ],
  "categories": [
    "math.CA",
    "math.PR"
  ],
  "abstract": "Let $\\mathbb F=\\mathbb R$ or $\\mathbb C$ and $n\\in\\b N$. Let $(S_k)_{k\\ge0}$ be a time-homogeneous random walk on $GL_n(\\b F)$ associated with an $U_n(\\b F)$-biinvariant measure $\\nu\\in M^1(GL_n(\\b F))$. We derive a central limit theorem for the ordered singular spectrum $\\sigma_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix. The main ingredient for the proof will be a oscillatory result for the spherical functions $\\phi_{i\\rho+\\lambda}$ of $(GL_n(\\b F),U_n(\\b F))$. More precisely, we present a necessarily unique mapping $m_{\\bf 1}:G\\to\\b R^n$ such that for some constant $C$ and all $g\\in G$, $\\lambda\\in\\b R^n$, $$|\\phi_{i\\rho+\\lambda}(g)- e^{i\\lambda\\cdot m_{\\bf 1}(g)}|\\le C\\|\\lambda\\|^2.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-22T10:12:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A90",
      "33C67",
      "22E46",
      "60B15",
      "60F05",
      "43A62"
    ],
    "keywords": [
      "central limit theorem",
      "uniform oscillatory behavior",
      "spherical functions",
      "explicit analytic formulas",
      "ordered singular spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.4866V"
    }
  }
}