 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.