# Glivenko-Cantelli lemma

Let ${X}_{1},\mathrm{\dots},{X}_{n}$ be iid as $X$ with (unknown) distribution function^{} $F$. Let $\omega $ be the outcome and ${F}_{n}(x,\omega )$ be the empirical distribution function based on observations ${X}_{1}(\omega ),\mathrm{\dots},{X}_{n}(\omega )$. Then, as $n\to \mathrm{\infty}$,

$$ |

where a.s. means almost surely.

Title | Glivenko-Cantelli lemma |
---|---|

Canonical name | GlivenkoCantelliLemma |

Date of creation | 2013-03-22 14:33:24 |

Last modified on | 2013-03-22 14:33:24 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 4 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 62G20 |