{
  "id": "2408.10068",
  "version": "v1",
  "published": "2024-08-19T15:09:12.000Z",
  "updated": "2024-08-19T15:09:12.000Z",
  "title": "Analysis of The Limiting Spectral Distribution of Large Random Matrices of The Marčenko-Pastur Type",
  "authors": [
    "Haoran Li"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Consider the random matrix \\(\\bW_n = \\bB_n + n^{-1}\\bX_n^*\\bA_n\\bX_n\\), where \\(\\bA_n\\) and \\(\\bB_n\\) are Hermitian matrices of dimensions \\(p \\times p\\) and \\(n \\times n\\), respectively, and \\(\\bX_n\\) is a \\(p \\times n\\) random matrix with independent and identically distributed entries of mean 0 and variance 1. Assume that \\(p\\) and \\(n\\) grow to infinity proportionally, and that the spectral measures of \\(\\bA_n\\) and \\(\\bB_n\\) converge as \\(p, n \\to \\infty\\) towards two probability measures \\(\\calA\\) and \\(\\calB\\). Building on the groundbreaking work of \\cite{marchenko1967distribution}, which demonstrated that the empirical spectral distribution of \\(\\bW_n\\) converges towards a probability measure \\(F\\) characterized by its Stieltjes transform, this paper investigates the properties of \\(F\\) when \\(\\calB\\) is a general measure. We show that \\(F\\) has an analytic density at the region near where the Stieltjes transform of $\\calB$ is bounded. The density closely resembles \\(C\\sqrt{|x - x_0|}\\) near certain edge points \\(x_0\\) of its support for a wide class of \\(\\calA\\) and \\(\\calB\\). We provide a complete characterization of the support of \\(F\\). Moreover, we show that \\(F\\) can exhibit discontinuities at points where \\(\\calB\\) is discontinuous.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-19T15:09:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52"
    ],
    "keywords": [
      "large random matrices",
      "random matrix",
      "limiting spectral distribution",
      "marčenko-pastur type",
      "probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}