{
  "id": "2103.03204",
  "version": "v1",
  "published": "2021-03-04T18:14:44.000Z",
  "updated": "2021-03-04T18:14:44.000Z",
  "title": "On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations",
  "authors": [
    "Alicja Dembczak-Kołodziejczyk",
    "Anna Lytova"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Given $n,m\\in \\mathbb{N}$, we study two classes of large random matrices of the form $$ \\mathcal{L}_n =\\sum_{\\alpha=1}^m\\xi_\\alpha \\mathbf{y}_\\alpha \\mathbf{y}_\\alpha ^T\\quad\\text{and}\\quad \\mathcal{A}_n =\\sum_{\\alpha =1}^m\\xi_\\alpha (\\mathbf{y}_\\alpha \\mathbf{x}_\\alpha ^T+\\mathbf{x}_\\alpha \\mathbf{y}_\\alpha ^T), $$ where for every $n$, $(\\xi_\\alpha )_\\alpha \\subset \\mathbb{R}$ are iid random variables independent of $(\\mathbf{x}_\\alpha,\\mathbf{y}_\\alpha)_\\alpha$, and $(\\mathbf{x}_\\alpha )_\\alpha $, $(\\mathbf{y}_\\alpha )_\\alpha \\subset \\mathbb{R}^n$ are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as $n,m(n)\\to \\infty$: a standard one, where $m/n\\to c$, and a slightly modified one, where $m/n\\to\\infty$ and $\\mathbf{E}\\xi\\to 0$ while $m\\mathbf{E}\\xi /n\\to c$ for some $c\\ge 0$. Assuming that vectors $(\\mathbf{x}_\\alpha )_\\alpha $ and $(\\mathbf{y}_\\alpha )_\\alpha $ are normalized and isotropic \"in average\", we prove the convergence in probability of the empirical spectral distributions of $\\mathcal{L}_n $ and $\\mathcal{A}_n $ to a version of the Marchenko-Pastur law and so called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as $(\\xi_\\alpha )_\\alpha $, in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [9, 21].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-04T18:14:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60B20",
      "47N30"
    ],
    "keywords": [
      "empirical spectral distribution",
      "sample covariance matrices",
      "normalized rademacher random variables",
      "iid random variables independent",
      "correlations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}