{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[i.i.d.|00MRN0FARV]] real random variables with [[cumulative distribution function|0H8H7ZTHW2]] $F$.\nFor $n \\ge 1$ and $x \\in \\R$, let\n$$F_n(x) = \\frac{1}{n} \\#\\set{k \\le n : X_k \\le x}$$\nbe the empirical distribution function.\nThen, [[almost surely|0ST21PGDAP]],\n$$\\sup_{x \\in \\R} \\abs{F_n(x) - F(x)} \\to 0 \\quad \\text{as } n \\to \\infty.$$\n\n## Remarks\n\nFor each fixed $x$, the [[strong law of large numbers|0DFWHQ839T]] applied to the indicators $\\mathbf{1}_{\\set{X_k \\le x}}$ gives $F_n(x) \\to F(x)$ almost surely.\nThe same holds for the left limits $F_n(x-) \\to F(x-)$.\nTo make the convergence [[uniform|08VZ57PT26]], choose finitely many points $x_0 \u003c x_1 \u003c \\cdots \u003c x_m$ such that $F$ grows by at most $\\varepsilon$ between consecutive points, apart from jumps at the points themselves.\nSince $F_n$ and $F$ are non-decreasing, the uniform error is at most $\\varepsilon$ plus the error at these finitely many points.\n\nExample: if the $X_k$ are uniform on $[0, 1]$, then the step function $F_n$ lies within any fixed distance of the line $F(x) = x$ on $[0, 1]$, for all large $n$, almost surely.\n\nThe rate is given by the Dvoretzky–Kiefer–Wolfowitz inequality, and the fluctuations of order $n^{-1/2}$ are described by the [[empirical process|XKFG1676CG]] and [[Donsker's theorem|818832PMSV]].\n\n## Explanations\n\n### L1 human\n\nTake a large sample from an unknown distribution and draw the step function that records which fraction of the sample lies below each value.\nThis step function gets close to the true distribution function everywhere at once as the sample grows.\nSo a large sample reveals the whole distribution, not only its mean.\n\n### L2 human\n\nThe theorem is often called the fundamental theorem of statistics, since it justifies estimating a distribution by its empirical version.\nIt is the first uniform law of large numbers.\nEmpirical process theory generalises it to classes of sets or functions, called Glivenko–Cantelli classes, on which the empirical averages converge uniformly; the Vapnik–Chervonenkis condition is a sufficient criterion.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "0H8H7ZTHW2"
    ],
    "kind": "statement",
    "msc": [
      "62G30",
      "60F15"
    ],
    "name": "Glivenko–Cantelli theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q544369",
      "wikipedia": "Glivenko–Cantelli theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem"
      }
    ],
    "tag": "E7B78X780X",
    "uses": [
      "00MRN0FARV",
      "0H8H7ZTHW2",
      "0ST21PGDAP",
      "08VZ57PT26"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/E7B78X780X/",
  "links": [
    {
      "field": "about",
      "target": "0H8H7ZTHW2"
    },
    {
      "field": "uses",
      "target": "00MRN0FARV"
    },
    {
      "field": "uses",
      "target": "0H8H7ZTHW2"
    },
    {
      "field": "uses",
      "target": "0ST21PGDAP"
    },
    {
      "field": "uses",
      "target": "08VZ57PT26"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/E7B78X780X.md",
  "msc": [
    "62G30",
    "60F15"
  ],
  "name": "Glivenko–Cantelli theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/E7B/E7B78X780X.md",
  "status": null,
  "tag": "E7B78X780X",
  "uri": "m2://E7B78X780X",
  "url": "https://math2.tamnd.com/wiki/glivenko-cantelli-theorem/",
  "writing": {
    "tag": "E7B78X780X",
    "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
      }
    ]
  }
}