{
  "id": "1108.5464",
  "version": "v2",
  "published": "2011-08-27T16:52:27.000Z",
  "updated": "2012-10-29T22:20:07.000Z",
  "title": "Limit Theory for the largest eigenvalues of sample covariance matrices with heavy-tails",
  "authors": [
    "Richard A. Davis",
    "Oliver Pfaffel",
    "Robert Stelzer"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We study the joint limit distribution of the $k$ largest eigenvalues of a $p\\times p$ sample covariance matrix $XX^\\T$ based on a large $p\\times n$ matrix $X$. The rows of $X$ are given by independent copies of a linear process, $X_{it}=\\sum_j c_j Z_{i,t-j}$, with regularly varying noise $(Z_{it})$ with tail index $\\alpha\\in(0,4)$. It is shown that a point process based on the eigenvalues of $XX^\\T$ converges, as $n\\to\\infty$ and $p\\to\\infty$ at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on $\\alpha$ and $\\sum c_j^2$. This result is extended to random coefficient models where the coefficients of the linear processes $(X_{it})$ are given by $c_j(\\theta_i)$, for some ergodic sequence $(\\theta_i)$, and thus vary in each row of $X$. As a by-product of our techniques we obtain a proof of the corresponding result for matrices with iid entries in cases where $p/n$ goes to zero or infinity and $\\alpha\\in(0,2)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-10-29T22:20:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "62G32"
    ],
    "keywords": [
      "sample covariance matrices",
      "largest eigenvalues",
      "limit theory",
      "heavy-tails",
      "joint limit distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.5464D"
    }
  }
}