{
  "id": "1310.8094",
  "version": "v2",
  "published": "2013-10-30T10:36:33.000Z",
  "updated": "2015-02-04T15:08:53.000Z",
  "title": "Analysis of the limiting spectral measure of large random matrices of the separable covariance type",
  "authors": [
    "Romain Couillet",
    "Walid Hachem"
  ],
  "comment": "Correction of the proof of Lemma 3.3",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the random matrix $\\Sigma = D^{1/2} X \\widetilde D^{1/2}$ where $D$ and $\\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions $N \\times N$ and $n \\times n$, and where $X$ is a random matrix with independent and identically distributed centered elements with variance $1/n$. Assume that the dimensions $N$ and $n$ grow to infinity at the same pace, and that the spectral measures of $D$ and $\\widetilde D$ converge as $N,n \\to\\infty$ towards two probability measures. Then it is known that the spectral measure of $\\Sigma\\Sigma^*$ converges towards a probability measure $\\mu$ characterized by its Stieltjes Transform. In this paper, it is shown that $\\mu$ has a density away from zero, this density is analytical wherever it is positive, and it behaves in most cases as $\\sqrt{|x - a|}$ near an edge $a$ of its support. A complete characterization of the support of $\\mu$ is also provided. \\\\ Beside its mathematical interest, this analysis finds applications in a certain class of statistical estimation problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-30T10:36:33.000Z",
      "abstract": "Consider the random matrix $\\Sigma = D^{1/2} X \\widetilde D^{1/2}$ where $D$ and $\\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions $N \\times N$ and $n \\times n$, and where $X$ is a random matrix with independent and identically distributed centered elements with variance $1/n$. Assume that the dimensions $N$ and $n$ grow to infinity at the same pace, and that the spectral measures of $D$ and $\\widetilde D$ converge as $N,n \\to\\infty$ towards two probability measures. Then it is known that the spectral measure of $\\Sigma\\Sigma^*$ converges towards a probability measure $\\mu$ characterized by its Stieltjes Transform. \\\\ In this paper, it is shown that $\\mu$ has a density away from zero, this density is analytical wherever it is positive, and it behaves in most cases as $\\sqrt{|x - a|}$ near an edge $a$ of its support. A complete characterization of the support of $\\mu$ is also provided. \\\\ Beside its mathematical interest, this analysis finds applications in a certain class of statistical estimation problems.",
      "comment": "1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-04T15:08:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large random matrices",
      "limiting spectral measure",
      "separable covariance type",
      "random matrix",
      "probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.8094C"
    }
  }
}