= Complete convergence of random variables
= Complete convergence
{synonym}
A sequence converges completely to $X$ when $\sum_n\Pr(|X_n-X|>\epsilon)<\infty$ for every $\epsilon>0$. 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: $X_n=1_{\{U\le1/n\}}$ for one uniform random variable $U$ converges almost surely to zero but its exceedance probabilities have a divergent sum.
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