criterion for almost-sure convergence

Let $X_{1},X_{2},\ldots$ and $X$ be random variables. If, for every $\epsilon>0$ , the sum $\sum_{n=1}^{\infty}\mathbb{P}(\lvert X_{n}-X\rvert>\epsilon)$ is finite, then $X_{n}$ converge to $X$ almost surely.

Proof.

By the Borel-Cantelli lemma, we have $\mathbb{P}(\limsup_{n}\{\lvert X_{n}-X\rvert>\epsilon\})=0$ . But $\limsup_{n}\{\lvert X_{n}-X\rvert>\epsilon\}$ is the same as the event $\{\limsup_{n}\lvert X_{n}-X\rvert>\epsilon\}$ . (The latter event involves the limit superior of numbers (http://planetmath.org/LimitSuperior); the former involves the limit superior of sets (http://planetmath.org/InfinitelyOften).) So taking the limit $\epsilon\searrow 0$ , we have $\mathbb{P}(\limsup_{n}\lvert X_{n}-X\rvert>0)=0$ , or equivalently $\mathbb{P}(\limsup_{n}\lvert X_{n}-X\rvert=0)=1$ . ∎

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