convergence in distribution

A sequence of distribution functionsMathworldPlanetmath F1,F2, converges weakly to a distribution function F if Fn(t)F(t) for each point t at which F is continuous.

If the random variablesMathworldPlanetmath X,X1,X2, have associated distribution functions F,F1,F2,, respectively, then we say that Xn converges in distributionPlanetmathPlanetmath to X, and denote this by Xn𝐷X.

This definition holds for joint distribution functionsMathworldPlanetmath 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 theoremsMathworldPlanetmath, 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