BorelCantelli

Borel-Cantelli lemmas are two results in probability theory that describe when events occur infinitely often. Named for Émile Borel and Paolo Cantelli, they concern a sequence of events A1, A2, … in a probability space and the almost sure behavior of the event that A_n occurs infinitely often (denoted A_n i.o.).

The first Borel-Cantelli lemma states that if the sum of the probabilities of A_n is finite, sum_{n=1}^∞ P(A_n) < ∞, then the probability that A_n occurs infinitely often is zero: P(A_n i.o.) = 0. Informally, only finitely many A_n occur with probability one.

The second Borel-Cantelli lemma provides a converse under independence: if the events A_n are independent and the sum of their probabilities diverges, sum_{n=1}^∞ P(A_n) = ∞, then P(A_n i.o.) = 1; i.e., with probability one infinitely many A_n occur. Independence is essential in this form.

Without independence, the second result may fail, but several generalizations exist. Variants hold under weaker hypotheses such as certain quasi-independence conditions or using more advanced criteria (e.g., Kochen–Stone type results) that relate long-term occurrence to correlations among the events.

Borel-Cantelli lemmas are used throughout probability theory and related fields, including ergodic theory, number theory, and statistics, to deduce almost sure statements about sequences of random events, such as convergence properties and the frequency of rare events.