{
  "id": "math/0307330",
  "version": "v3",
  "published": "2003-07-25T04:26:37.000Z",
  "updated": "2006-02-27T10:18:42.000Z",
  "title": "Spectral measure of large random Hankel, Markov and Toeplitz matrices",
  "authors": [
    "Włodzimierz Bryc",
    "Amir Dembo",
    "Tiefeng Jiang"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117905000000495 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 1, 1-38",
  "doi": "10.1214/009117905000000495",
  "categories": [
    "math.PR",
    "math.CO",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables $\\{X_k\\}$ of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables $\\{X_{ij}\\}_{j>i}$ of zero mean and unit variance, scaling the eigenvalues by $\\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions $\\gamma_H$, $\\gamma_M$ and $\\gamma_T$ of unbounded support. The moments of $\\gamma_H$ and $\\gamma_T$ are the sum of volumes of solids related to Eulerian numbers, whereas $\\gamma_M$ has a bounded smooth density given by the free convolution of the semicircle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables $\\{X_{ij}\\}_{j>i}$ of mean $m$ and finite variance, scaling the eigenvalues by ${n}$ we prove the almost sure, weak convergence of the spectral measures to the atomic measure at $-m$. If $m=0$, and the fourth moment is finite, we prove that the spectral norm of $\\mathbf {M}_n$ scaled by $\\sqrt{2n\\log n}$ converges almost surely to 1.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-02-27T10:18:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "60F99",
      "62H10",
      "60F10"
    ],
    "keywords": [
      "spectral measure",
      "large random hankel",
      "toeplitz matrices",
      "symmetric markov matrices",
      "random variables"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......7330B"
    }
  }
}