Borel–Cantelli lemma

The Borel–Cantelli lemmas relate the probability that infinitely many events occur to the convergence or divergence of the sum of their probabilities. The first requires no independence; the converse holds when the events are independent.

Limit superior of events: The event that infinitely many events in a sequence occur. This is the event whose probability the lemmas bound or determine.
Probability space: A mathematical structure consisting of outcomes, measurable events, and a probability measure. The events and probabilities in the theorem live in this structure.
Strong law of large numbers: A theorem asserting almost-sure convergence of sample averages to an expected value under specified conditions. Borel–Cantelli often converts tail bounds into almost-sure convergence in its proofs.
Émile Borel: A French mathematician whose work helped establish measure theory and modern probability. The first lemma bears his name and grew from his work on probability.
First Borel–Cantelli lemma: A theorem stating that summable event probabilities imply probability zero of infinitely many occurrences. It gives the convergence direction without requiring independence.
Event (probability theory): A measurable subset of a probability space, interpreted as a possible outcome condition. The theorem applies to sequences of such measurable sets.
Convergence of random variables: The study of modes in which sequences of random variables approach limiting random variables. The lemmas provide a standard route from probability bounds to almost-sure convergence.
Francesco Paolo Cantelli: An Italian mathematician known for foundational contributions to probability theory. His work established the independent-events converse commonly called the second lemma.
Second Borel–Cantelli lemma: A theorem stating that independent events with divergent probability sum occur infinitely often with probability one. It supplies the converse under independence.
Countable subadditivity: The property that a countable union has probability no greater than the sum of its probabilities. It controls probabilities of unions of all events after a given index.

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