# convergence in distribution

A sequence of distribution functions^{} ${F}_{1},{F}_{2},\mathrm{\dots}$ converges *weakly* to a distribution
function $F$ if ${F}_{n}(t)\to F(t)$ for each point $t$ at which $F$ is continuous.

If the random variables^{} $X,{X}_{1},{X}_{2},\mathrm{\dots}$ have associated distribution functions
$F,{F}_{1},{F}_{2},\mathrm{\dots}$, respectively, then we say that ${X}_{n}$ converges *in distribution ^{}* to
$X$, and denote this by ${X}_{n}\stackrel{\mathit{D}}{\to}X$.

This definition holds for joint distribution functions^{} and random vectors as well.

This is probably the weakest of convergence of random variables. Some results involving this of convergence
are the central limit theorems^{}, Helly-Bray theorem, Paul Lévy continuity theorem, Cramér-Wold theorem and Scheffé’s theorem.

Title | convergence in distribution |
---|---|

Canonical name | ConvergenceInDistribution |

Date of creation | 2013-03-22 13:14:12 |

Last modified on | 2013-03-22 13:14:12 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 11 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 60E05 |

Related topic | WeakConvergence |