criterion for almost-sure convergence

Let X1,X2, and X be random variablesMathworldPlanetmath. If, for every ϵ>0, the sum n=1(|Xn-X|>ϵ) is finite, then Xn converge to X almost surely.


By the Borel-Cantelli lemmaMathworldPlanetmath, we have (lim supn{|Xn-X|>ϵ})=0. But lim supn{|Xn-X|>ϵ} is the same as the event {lim supn|Xn-X|>ϵ}. (The latter event involves the limit superior of numbers (; the former involves the limit superior of sets ( So taking the limit ϵ0, we have (lim supn|Xn-X|>0)=0, or equivalently (lim supn|Xn-X|=0)=1. ∎

Title criterion for almost-sure convergence
Canonical name CriterionForAlmostsureConvergence
Date of creation 2013-03-22 15:54:45
Last modified on 2013-03-22 15:54:45
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 15
Author stevecheng (10074)
Entry type Corollary
Classification msc 60A99
Synonym corollary of Borel-Cantelli lemma