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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All labels observed (4)

Label	Occurrences
Borel–Cantelli lemma	3
Borel–Cantelli lemmas canonical	2
Borel–Cantelli lemma for dependent events	1
Borel’s strong law of large numbers formulation	1

How this entity was disambiguated

This entity first appeared as the object of triple T13444054 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Borel–Cantelli lemmas
Context triple: [Kolmogorov zero–one law, relatedTo, Borel–Cantelli lemmas]

A. Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
B. Khinchin's law of the iterated logarithm
Khinchin's law of the iterated logarithm is a fundamental result in probability theory that precisely characterizes the almost-sure fluctuations of partial sums of independent random variables on the scale of the square root of twice the product of their variance and the iterated logarithm of the sample size.
C. Erdős–Rényi law of large numbers
The Erdős–Rényi law of large numbers is a refinement of the classical law of large numbers that provides precise asymptotic behavior and convergence rates for sums of independent random variables, developed by mathematicians Pál Erdős and Alfréd Rényi.
D. Limit Laws for Sums of Independent Random Variables
Limit Laws for Sums of Independent Random Variables is a foundational mathematical work that systematically develops the theory of probability limit theorems, including results such as the law of large numbers and central limit behavior for sums of independent random variables.
E. Cramér’s theorem in large deviations
Cramér’s theorem in large deviations is a fundamental result in probability theory that characterizes the exponential decay rate of tail probabilities for sums of independent, identically distributed random variables via a convex rate function.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Borel–Cantelli lemmas
Target entity description: 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.

A. Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
B. Khinchin's law of the iterated logarithm
Khinchin's law of the iterated logarithm is a fundamental result in probability theory that precisely characterizes the almost-sure fluctuations of partial sums of independent random variables on the scale of the square root of twice the product of their variance and the iterated logarithm of the sample size.
C. Erdős–Rényi law of large numbers
The Erdős–Rényi law of large numbers is a refinement of the classical law of large numbers that provides precise asymptotic behavior and convergence rates for sums of independent random variables, developed by mathematicians Pál Erdős and Alfréd Rényi.
D. Limit Laws for Sums of Independent Random Variables
Limit Laws for Sums of Independent Random Variables is a foundational mathematical work that systematically develops the theory of probability limit theorems, including results such as the law of large numbers and central limit behavior for sums of independent random variables.
E. Cramér’s theorem in large deviations
Cramér’s theorem in large deviations is a fundamental result in probability theory that characterizes the exponential decay rate of tail probabilities for sums of independent, identically distributed random variables via a convex rate function.
F. None of above. chosen

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 ⓘ
hasGeneralization	Borel–Cantelli lemma for dependent events ⓘ linked to: Borel–Cantelli lemmas
hasPart	first Borel–Cantelli lemma ⓘ second Borel–Cantelli lemma ⓘ
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 ⓘ Émile Borel ⓘ
relatedTo	Kolmogorov zero–one law ⓘ almost sure convergence of random variables ⓘ law of the iterated logarithm ⓘ linked to: Kolmogorov's law of the iterated logarithm 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 ⓘ

How these facts were elicited