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 $X_n\stackrel{𝐷}{\to }X$.

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