{
  "id": "2003.03857",
  "version": "v1",
  "published": "2020-03-08T22:05:36.000Z",
  "updated": "2020-03-08T22:05:36.000Z",
  "title": "Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations",
  "authors": [
    "Johannes Heiny",
    "Jianfeng Yao"
  ],
  "comment": "30 pages, 6 figures",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Consider a $p$-dimensional population ${\\mathbf x} \\in\\mathbb{R}^p$ with iid coordinates in the domain of attraction of a stable distribution with index $\\alpha\\in (0,2)$. Since the variance of ${\\mathbf x}$ is infinite, the sample covariance matrix ${\\mathbf S}_n=n^{-1}\\sum_{i=1}^n {{\\mathbf x}_i}{\\mathbf x}'_i$ based on a sample ${\\mathbf x}_1,\\ldots,{\\mathbf x}_n$ from the population is not well behaved and it is of interest to use instead the sample correlation matrix ${\\mathbf R}_n= \\{\\operatorname{diag}({\\mathbf S}_n)\\}^{-1/2}\\, {\\mathbf S}_n \\{\\operatorname{diag}({\\mathbf S}_n)\\}^{-1/2}$. This paper finds the limiting distributions of the eigenvalues of ${\\mathbf R}_n$ when both the dimension $p$ and the sample size $n$ grow to infinity such that $p/n\\to \\gamma \\in (0,\\infty)$. The family of limiting distributions $\\{H_{\\alpha,\\gamma}\\}$ is new and depends on the two parameters $\\alpha$ and $\\gamma$. The moments of $H_{\\alpha,\\gamma}$ are fully identified as sum of two contributions: the first from the classical Mar\\v{c}enko-Pastur law and a second due to heavy tails. Moreover, the family $\\{H_{\\alpha,\\gamma}\\}$ has continuous extensions at the boundaries $\\alpha=2$ and $\\alpha=0$ leading to the Mar\\v{c}enko-Pastur law and a modified Poisson distribution, respectively. Our proofs use the method of moments, the path-shortening algorithm developed in [18] and some novel graph counting combinatorics. As a consequence, the moments of $H_{\\alpha,\\gamma}$ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions $H_{\\alpha,\\gamma}$ is also provided for comparison with the Mar\\v{c}enko-Pastur law.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-08T22:05:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "60F05",
      "60F10",
      "60G10",
      "60G55",
      "60G70"
    ],
    "keywords": [
      "sample correlation matrix",
      "limiting distributions",
      "heavy-tailed populations",
      "eigenvalues",
      "sample covariance matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}