{
  "id": "math/0607767",
  "version": "v4",
  "published": "2006-07-29T17:21:35.000Z",
  "updated": "2007-10-17T18:02:34.000Z",
  "title": "Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles",
  "authors": [
    "Alain Rouault"
  ],
  "comment": "51 pages ; it replaces and extends arXiv:math/0607767 and arXiv:math/0509021 Third version: corrected constants in Theorem 3.1",
  "journal": "Latin American Journal of Probability and Mathematical Statistics (ALEA) 3 (2007) 181-230",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If $n$ is the size of the sample, $r\\leq n$ the number of variates and $X_{n,r}$ such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of $\\det X_{n,r}$ into a product of $r$ independent gamma or beta random variables. For $n$ fixed, we study the evolution as $r$ grows, and then take the limit of large $r$ and $n$ with $r/n = t \\leq 1$. We derive limit theorems for the sequence of {\\it processes with independent increments} $\\{n^{-1} \\log \\det X_{n, \\lfloor nt\\rfloor}, t \\in [0, T]\\}_n$ for $T \\leq 1$.. Since the logarithm of the determinant is a linear statistic of the empirical spectral distribution, we connect the results for marginals (fixed $t$) with those obtained by the spectral method. Actually, all the results hold true for $\\beta$ models, if we define the determinant as the product of charges.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-10-17T18:02:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behavior",
      "random determinants",
      "jacobi ensemble",
      "sample covariance matrices",
      "sample correlation matrices"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7767R"
    }
  }
}