Complete convergence of random variables

A sequence converges completely to $X$ when ${\sum }_n\mathrm{Pr}(|X_n-X|>ϵ)<\infty$ for every $ϵ>0$. The first Borel-Cantelli lemma implies almost sure convergence. For independent $X_n$, any limit in probability is almost surely constant; the second Borel-Cantelli lemma then gives the converse from almost sure to complete convergence. Independence is essential: $X_n=1_{\{U\le 1/n\}}$ for one uniform random variable $U$ converges almost surely to zero but its exceedance probabilities have a divergent sum.

If independent random variables $X_n$ converge almost surely to a finite $X$, then ${\mathrm{limsup}}_n{X}_n$ is measurable for the tail sigma-algebra. The Kolmogorov zero-one law forces every rational threshold event for that limit to have probability zero or one. A real-valued probability distribution with this property is a point mass, so $X$ is almost surely constant. Consequently the Second Borel-Cantelli lemma upgrades almost sure convergence of independent sequences to complete convergence.