The Borel-Cantelli Lemma gives conditions under which infinitely many events in a sequence occur with probability zero or with probability one, depending on whether the sum of their individual probabilities is finite or, under independence, infinite. Named for Emile Borel and Francesco Paolo Cantelli, it is a basic tool in probability theory for reasoning about the limiting behavior of infinite sequences of events.
Facts
StatementThe Borel-Cantelli lemma states, in its first part, that if the sum of the probabilities of a sequence of events is finite then the probability that infinitely many of them occur is zero, and in its second part, that if the events are independent and the sum of their probabilities is infinite then the probability that infinitely many of them occur is one. 1 Proof YearYear given is for the first part, formulated by Emile Borel in the course of his work on normal numbers; the independently proved second part (Cantelli) is not dated in the sources checked this pass and is left an open gap. Classification
Statement Form Connections
Sources
1. Borel-Cantelli Lemma (Wikipedia)
Wikimedia FoundationStatement of lemma for probability spaces sectionQuote, Statement of lemma for probability spaces section
The first Borel-Cantelli lemma, which states that if the sum of the probabilities of the events is finite, then the probability that infinitely many of them occur is 0.
2. Emile Borel (Wikipedia)
Wikimedia Foundationprobability and randomness discussion, 1909 normal numbers passageQuote, probability and randomness discussion, 1909 normal numbers passage
In 1909, Borel formulated the notion that numbers picked randomly on the basis of their value are almost always normal, and with explicit constructions in terms of digits, it is quite straightforward to get numbers that are normal.
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