{
  "id": "1012.2710",
  "version": "v3",
  "published": "2010-12-13T12:32:42.000Z",
  "updated": "2011-04-26T15:38:08.000Z",
  "title": "On the Asymptotic Spectrum of Products of Independent Random Matrices",
  "authors": [
    "Friedrich Götze",
    "Alexander Tikhomirov"
  ],
  "comment": "Complete formulas (4.14); correct typos",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider products of independent random matrices with independent entries. The limit distribution of the expected empirical distribution of eigenvalues of such products is computed. Let $X^{(\\nu)}_{jk},{}1\\le j,r\\le n$, $\\nu=1,...,m$ be mutually independent complex random variables with $\\E X^{(\\nu)}_{jk}=0$ and $\\E {|X^{(\\nu)}_{jk}|}^2=1$. Let $\\mathbf X^{(\\nu)}$ denote an $n\\times n$ matrix with entries $[\\mathbf X^{(\\nu)}]_{jk}=\\frac1{\\sqrt{n}}X^{(\\nu)}_{jk}$, for $1\\le j,k\\le n$. Denote by $\\lambda_1,...,\\lambda_n$ the eigenvalues of the random matrix $\\mathbf W:= \\prod_{\\nu=1}^m\\mathbf X^{(\\nu)}$ and define its empirical spectral distribution by $$ \\mathcal F_n(x,y)=\\frac1n\\sum_{k=1}^n\\mathbb I\\{\\re{\\lambda_k}\\le x,\\im{\\lambda_k\\le y}\\}, $$ where $\\mathbb I\\{B\\}$ denotes the indicator of an event $B$. We prove that the expected spectral distribution $F_n^{(m)}(x,y)=\\E \\mathcal F_n^{(m)}(x,y)$ converges to the distribution function $G(x,y)$ corresponding to the $m$-th power of the uniform distribution on the unit disc in the plane $\\mathbb R^2$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-04-26T15:38:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60B20",
      "15B52"
    ],
    "keywords": [
      "independent random matrices",
      "asymptotic spectrum",
      "spectral distribution",
      "mutually independent complex random variables",
      "eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.2710G"
    }
  }
}