Borel–Cantelli lemmas

E1041768

mathematical lemma probability theory theorem

The Borel–Cantelli lemmas are fundamental results in probability theory that characterize when events occur infinitely often or only finitely often, based on the convergence or divergence of the sum of their probabilities.

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Observed surface forms (4)

Surface form	Occurrences
Borel–Cantelli lemma	1
Borel’s strong law of large numbers formulation	1
first Borel–Cantelli lemma	0
second Borel–Cantelli lemma	0

Statements (46)

Predicate	Object
instanceOf	mathematical lemma ⓘ probability theory theorem ⓘ probability theory theorem ⓘ probability theory theorem ⓘ
appearsIn	graduate-level probability textbooks ⓘ
appliesTo	countable sequences of events ⓘ
assumption	no independence assumption required ⓘ pairwise independence or independence of events ⓘ
characterizes	conditions for events to occur infinitely often ⓘ conditions for events to occur only finitely often ⓘ
concerns	almost sure behavior of events ⓘ divergence of sum of probabilities ⓘ events occurring infinitely often ⓘ events occurring only finitely often ⓘ sequences of events in probability spaces ⓘ sum of probabilities of events ⓘ
conclusionType	almost sure statement ⓘ
field	probability theory ⓘ
formalizedIn	measure-theoretic probability ⓘ
generalizedBy	Kochen–Stone theorem NERFINISHED ⓘ
hasGeneralization	Borel–Cantelli lemma for dependent events NERFINISHED ⓘ
hasPart	first Borel–Cantelli lemma ⓘ second Borel–Cantelli lemma NERFINISHED ⓘ
implies	zero-one law for limsup of events under independence ⓘ
importance	fundamental tool in proving almost sure convergence results ⓘ
logicalForm	implication between series of probabilities and limsup event probability ⓘ
namedAfter	Francesco Paolo Cantelli NERFINISHED ⓘ Émile Borel NERFINISHED ⓘ
relatedTo	Kolmogorov zero–one law NERFINISHED ⓘ almost sure convergence of random variables ⓘ law of the iterated logarithm NERFINISHED ⓘ strong law of large numbers ⓘ
states	if events are independent and the sum of their probabilities diverges then the probability that infinitely many occur is one ⓘ if the sum of probabilities of events is finite then the probability that infinitely many of them occur is zero ⓘ
typicalCondition	sum_{n=1}^∞ P(A_n) < ∞ ⓘ sum_{n=1}^∞ P(A_n) = ∞ ⓘ
typicalNotation	limsup A_n ⓘ
usedIn	ergodic theory ⓘ limit theorems in probability ⓘ number theory ⓘ random series analysis ⓘ
usesConcept	almost sure convergence ⓘ independence of events ⓘ limsup of events ⓘ probability measure ⓘ sigma-algebra ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kolmogorov zero–one law → relatedTo → Borel–Cantelli lemmas ⓘ

Émile Borel → notableWork → Borel–Cantelli lemmas ⓘ

this entity surface form: Borel–Cantelli lemma

Émile Borel → notableWork → Borel–Cantelli lemmas ⓘ

this entity surface form: Borel’s strong law of large numbers formulation

Khinchin–Kolmogorov theorem → relatedTo → Borel–Cantelli lemmas ⓘ