{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(Y_{ij})_{i, j \\ge 1}$ be [[independent and identically distributed|00MRN0FARV]] real random variables with mean $0$ and variance $1$.\nLet $p = p(n)$ be positive integers with $p / n \\to y \\in (0, \\infty)$ as $n \\to \\infty$.\nLet $X_n$ be the $p \\times n$ matrix $(Y_{ij})_{i \\le p, j \\le n}$, let $S_n = \\frac1n X_n X_n^{\\mathsf{T}}$, and let $\\mu_n = \\frac1p \\sum_{k=1}^p \\delta_{\\lambda_k}$, where $\\lambda_1, \\ldots, \\lambda_p$ are the [[eigenvalues|0BQAA2BK3Q]] of $S_n$.\nLet $a = (1 - \\sqrt y)^2$ and $b = (1 + \\sqrt y)^2$.\nThen, almost surely, $\\mu_n$ [[converges weakly|0SRYJGXKR4]] to the Marchenko–Pastur distribution with ratio $y$, the probability measure\n$$\\nu_y(dx) = \\frac{1}{2\\pi x y} \\sqrt{(b - x)(x - a)} \\, \\mathbf{1}_{[a, b]}(x) \\, dx + \\Big(1 - \\frac1y\\Big)^{+} \\delta_0(dx),$$\nwhere $t^{+} = \\max(t, 0)$.\n\n## Remarks\n\nThe proof can use the moment method or the Stieltjes transform.\nThe Stieltjes transform $m(z) = \\int \\frac{\\nu_y(dx)}{x - z}$ of the limit solves the quadratic equation $y z m^2 + (z + y - 1) m + 1 = 0$, and inverting it gives the density.\nWhen $y \u003e 1$ the matrix $S_n$ has rank at most $n \u003c p$, which explains the atom of mass $1 - 1/y$ at $0$.\nThe limit has mean $1$ and second moment $1 + y$.\nExample: for $y = 1$ the support is $[0, 4]$ and the density is $\\frac{1}{2\\pi} \\sqrt{(4 - x)/x}$.\n\n## Explanations\n\n### L1 human\n\nTake $n$ independent samples of a vector of $p$ independent standard measurements and form the sample covariance matrix.\nIf $p$ is small compared with $n$, its eigenvalues are all close to $1$, the true value.\nIf $p$ and $n$ are both large and comparable, the eigenvalues spread over an interval around $1$ with a definite shape, the Marchenko–Pastur law.\n\n### L2 human\n\nIt is the analogue of the [[Wigner semicircle law|3QTT95DMFH]] for sample covariance matrices.\nIn high-dimensional statistics it shows that the sample covariance matrix is a poor estimate of the true covariance when $p/n$ is not small.\nEigenvalues of data that fall outside $[a, b]$ are read as signals that are not pure noise.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "0BQAA2BK3Q"
    ],
    "kind": "statement",
    "msc": [
      "60B20",
      "15B52",
      "62H10"
    ],
    "name": "Marchenko–Pastur law",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Marchenko–Pastur_distribution"
      }
    ],
    "tag": "DCK63WTY5E",
    "uses": [
      "0BQAA2BK3Q",
      "00MRN0FARV",
      "0SRYJGXKR4"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/DCK63WTY5E/",
  "links": [
    {
      "field": "about",
      "target": "0BQAA2BK3Q"
    },
    {
      "field": "uses",
      "target": "0BQAA2BK3Q"
    },
    {
      "field": "uses",
      "target": "00MRN0FARV"
    },
    {
      "field": "uses",
      "target": "0SRYJGXKR4"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/DCK63WTY5E.md",
  "msc": [
    "60B20",
    "15B52",
    "62H10"
  ],
  "name": "Marchenko–Pastur law",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DCK/DCK63WTY5E.md",
  "status": null,
  "tag": "DCK63WTY5E",
  "uri": "m2://DCK63WTY5E",
  "url": "https://math2.tamnd.com/wiki/marchenko-pastur-law/",
  "writing": {
    "tag": "DCK63WTY5E",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}