= Borel-Cantelli lemmas
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= Borel-Cantelli lemma
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The <Borel-Cantelli lemmas> turn a <series> of <event> <probabilities> into a statement about an entire infinite sequence of observations. A <summable sequence> of <probabilities> forces only finitely many occurrences <almost surely>, regardless of <independence>. A divergent <series> forces infinitely many occurrences when the <events> are mutually <independent events>. The precise <event> being measured is the <limsup of events>.
Imagine alarms with <probabilities> $p_n=1/(n+1)^2$ or $p_n=1/(n+1)$, starting at $n=1$. Both individual <probabilities> tend to zero. Nevertheless, the first alarm sequence eventually stops <almost surely>, whereas the second keeps returning <almost surely> if its <events> are mutually <independent events>. Reusing one <uniform random variable> for every alarm instead produces the <shared-uniform counterexample to the second Borel-Cantelli lemma>. Thus the individual <probabilities> alone do not determine long-run recurrence.
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