{
  "id": "1012.2743",
  "version": "v1",
  "published": "2010-12-13T14:44:46.000Z",
  "updated": "2010-12-13T14:44:46.000Z",
  "title": "On the asymptotic distribution of the singular values of powers of random matrices",
  "authors": [
    "Nikita Alexeev",
    "Friedrich Götze",
    "Alexander Tikhomirov"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider powers of random matrices with independent entries. Let $X_{ij}, i,j\\ge 1$, be independent complex random variables with $\\E X_{ij}=0$ and $\\E |X_{ij}|^2=1$ and let $\\mathbf X$ denote an $n\\times n$ matrix with $[\\mathbf X]_{ij}=X_{ij}$, for $1\\le i, j\\le n$. Denote by $s_1^{(m)}\\ge...\\ge s_n^{(m)}$ the singular values of the random matrix $\\mathbf W:={n^{-\\frac m2}} \\mathbf X^m$ and define the empirical distribution of the squared singular values by $$ \\mathcal F_n^{(m)}(x)=\\frac1n\\sum_{k=1}^nI_{\\{{s_k^{(m)}}^2\\le x\\}}, $$ where $I_{\\{B\\}}$ denotes the indicator of an event $B$. We prove that under a Lindeberg condition for the fourth moment that the expected spectral distribution $F_n^{(m)}(x)=\\E \\mathcal F_n^{(m)}(x)$ converges to the distribution function $G^{(m)}(x)$ defined by its moments $$ \\alpha_k(m):=\\int_{\\mathbb R}x^k\\,d\\,G(x)=\\frac {1}{mk+1}\\binom{km+k}{k}. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-13T14:44:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "15B52",
      "60B20"
    ],
    "keywords": [
      "random matrices",
      "asymptotic distribution",
      "independent complex random variables",
      "independent entries",
      "random matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.2743A"
    }
  }
}