{
  "id": "1010.3394",
  "version": "v2",
  "published": "2010-10-17T04:04:29.000Z",
  "updated": "2010-11-08T18:13:13.000Z",
  "title": "Fluctuation of Eigenvalues for Random Toeplitz and Related Matrices",
  "authors": [
    "Dang-Zheng Liu",
    "Xin Sun",
    "Zheng-Dong Wang"
  ],
  "comment": "27 pages, corrected small gap in proof of Theorem 1.1, added remark 1.3",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,...,$ being independent real random variables such that \\be \\mathbb{E}[a_{j}]=0, \\ \\ \\mathbb{E}[|a_{j}|^{2}]=1 \\ \\ \\textrm{for}\\,\\ \\ j=0,1,2,...,\\ee (homogeneity of 4-th moments) \\be{\\kappa=\\mathbb{E}[|a_{j}|^{4}],}\\ee \\noindent and further (uniform boundedness)\\be\\sup\\limits_{j\\geq 0} \\mathbb{E}[|a_{j}|^{k}]=C_{k}<\\iy\\ \\ \\ \\textrm{for} \\ \\ \\ k\\geq 3.\\ee Under the assumption of $a_{0}\\equiv 0$, we prove a central limit theorem for linear statistics of eigenvalues for a fixed polynomial with degree $\\geq 2$. Without the assumption, the CLT can be easily modified to a possibly non-normal limit law. In a special case where $a_{j}$'s are Gaussian, the result has been obtained by Chatterjee for some test functions. Our derivation is based on a simple trace formula for Toeplitz matrices and fine combinatorial analysis. Our method can apply to other related random matrix models, including Hankel matrices and product of several Toeplitz matrices in a flavor of free probability theory etc. Since Toeplitz matrices are quite different from the Wigner and Wishart matrices, our results enrich this topic.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-11-08T18:13:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60B20"
    ],
    "keywords": [
      "random toeplitz",
      "related matrices",
      "eigenvalues",
      "fluctuation",
      "random symmetric toeplitz matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.3394L"
    }
  }
}