= 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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