{
  "id": "0904.4091",
  "version": "v1",
  "published": "2009-04-27T06:30:55.000Z",
  "updated": "2009-04-27T06:30:55.000Z",
  "title": "Some asymptotic properties of the spectrum of the Jacobi ensemble",
  "authors": [
    "Holger Dette",
    "Jan Nagel"
  ],
  "comment": "20 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \\cdot \\prod_{i < j} | \\lambda_i - \\lambda_j |^\\beta \\prod^n_{i=1} (2 - \\lambda_i)^a (2 + \\lambda_i)^b I_{(-2,2)} (\\lambda_i) $$ $(a, b > -1, \\beta > 0) $ a strong uniform approximation by the roots of the Jacobi polynomials is derived if the parameters $a, b,$ $\\beta$ depend on $n$ and $n \\to \\infty$. Roughly speaking, the eigenvalues can be uniformly approximated by roots of Jacobi polynomials with parameters $((2a+2)/\\beta -1, (2b+2)/\\beta-1)$, where the error is of order $\\{\\log n/(a+b) \\}^{1/4}$. These results are used to investigate the asymptotic properties of the corresponding spectral distribution if $n \\to \\infty$ and the parameters $a, b$ and $\\beta$ vary with $n$. We also discuss further applications in the context of multivariate random $F$-matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-27T06:30:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "15A52"
    ],
    "keywords": [
      "asymptotic properties",
      "jacobi ensemble",
      "jacobi polynomials",
      "strong uniform approximation",
      "parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}