{
  "id": "0809.4698",
  "version": "v2",
  "published": "2008-09-26T19:50:00.000Z",
  "updated": "2009-09-25T11:06:41.000Z",
  "title": "Central limit theorem for linear eigenvalue statistics of random matrices with independent entries",
  "authors": [
    "A. Lytova",
    "L. Pastur"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/09-AOP452 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2009, Vol. 37, No. 5, 1778-1840",
  "doi": "10.1214/09-AOP452",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider $n\\times n$ real symmetric and Hermitian Wigner random matrices $n^{-1/2}W$ with independent (modulo symmetry condition) entries and the (null) sample covariance matrices $n^{-1}X^*X$ with independent entries of $m\\times n$ matrix $X$. Assuming first that the 4th cumulant (excess) $\\kappa_4$ of entries of $W$ and $X$ is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as $n\\to\\infty$, $m\\to\\infty$, $m/n\\to c\\in[0,\\infty)$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class $\\mathbf{C}^5$). This is done by using a simple ``interpolation trick'' from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially $\\mathbb{C}^5$ test function. Here the variance of statistics contains an additional term proportional to $\\kappa_4$. The proofs of all limit theorems follow essentially the same scheme.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-09-25T11:06:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "60F05",
      "62H99"
    ],
    "keywords": [
      "central limit theorem",
      "linear eigenvalue statistics",
      "independent entries",
      "hermitian wigner random matrices",
      "test function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.4698L"
    }
  }
}