# 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 CriterionForAlmostsureConvergence 2013-03-22 15:54:45 2013-03-22 15:54:45 stevecheng (10074) stevecheng (10074) 15 stevecheng (10074) Corollary msc 60A99 corollary of Borel-Cantelli lemma