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}^∞
The second Borel-Cantelli lemma provides a converse under independence: if the events A_n are independent and
Without independence, the second result may fail, but several generalizations exist. Variants hold under weaker hypotheses
Borel-Cantelli lemmas are used throughout probability theory and related fields, including ergodic theory, number theory, and