{
  "id": "1002.4442",
  "version": "v1",
  "published": "2010-02-24T00:34:33.000Z",
  "updated": "2010-02-24T00:34:33.000Z",
  "title": "Asymptotic distribution of singular values of powers of random matrices",
  "authors": [
    "Nikita Alexeev",
    "Friedrich Götze",
    "Alexander Tikhomirov"
  ],
  "comment": "16 pages, 5 figures",
  "journal": "Lithuanian Mathematical Journal, Vol. 50, No. 2, 2010, pp. 121-132",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $x$ be a complex random variable such that ${\\E {x}=0}$, ${\\E |x|^2=1}$, ${\\E |x|^{4} < \\infty}$. Let $x_{ij}$, $i,j \\in \\{1,2,...\\}$ be independet copies of $x$. Let ${\\Xb=(N^{-1/2}x_{ij})}$, $1\\leq i,j \\leq N$ be a random matrix. Writing $\\Xb^*$ for the adjoint matrix of $\\Xb$, consider the product $\\Xb^m{\\Xb^*}^m$ with some $m \\in \\{1,2,...\\}$. The matrix $\\Xb^m{\\Xb^*}^m$ is Hermitian positive semi-definite. Let $\\lambda_1,\\lambda_2,...,\\lambda_N$ be eigenvalues of $\\Xb^m{\\Xb^*}^m$ (or squared singular values of the matrix $\\Xb^m$). In this paper we find the asymptotic distribution function \\[ G^{(m)}(x)=\\lim_{N\\to\\infty}\\E{F_N^{(m)}(x)} \\] of the empirical distribution function \\[ {F_N^{(m)}(x)} = N^{-1} \\sum_{k=1}^N {\\mathbb{I}{\\{\\lambda_k \\leq x\\}}}, \\] where $\\mathbb{I} \\{A\\}$ stands for the indicator function of event $A$. The moments of $G^{(m)}$ satisfy \\[ M^{(m)}_p=\\int_{\\mathbb{R}}{x^p dG^{(m)}(x)}=\\frac{1}{mp+1}\\binom{mp+p}{p}. \\] In Free Probability Theory $M^{(m)}_p$ are known as Fuss--Catalan numbers. With $m=1$ our result turns to a well known result of Marchenko--Pastur 1967.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-24T00:34:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "15B52"
    ],
    "keywords": [
      "random matrices",
      "asymptotic distribution function",
      "free probability theory",
      "independet copies",
      "squared singular values"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.4442A"
    }
  }
}