Math 2.0

Probability theory and stochastic processes Limit theorems Lemma

First Borel–Cantelli lemma

Builds on Boole's inequalityContinuity of probability measuresEvent

Statement

Let A1,A2,…A_1, A_2, \dots be events in a probability space (Ω,ℱ,P)(\Omega, \mathcal{F}, P) with ∑n=1∞P(An)<∞\sum_{n=1}^{\infty} P(A_n) < \infty. Then P(lim sup⁡n→∞An)=0,wherelim sup⁡n→∞An=⋂n=1∞⋃m=n∞Am.P\left(\limsup_{n \to \infty} A_n\right) = 0, \quad \text{where} \quad \limsup_{n \to \infty} A_n = \bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} A_m. So almost surely, only finitely many of the events AnA_n occur.

Remarks

The event lim sup⁡nAn\limsup_n A_n is the set of outcomes that lie in infinitely many AnA_n. It is written {An i.o.}\{A_n \text{ i.o.}\}, for "infinitely often". Proof: for each nn, P(lim sup⁡mAm)≤P(⋃m≥nAm)≤∑m≥nP(Am)P(\limsup_m A_m) \le P\left(\bigcup_{m \ge n} A_m\right) \le \sum_{m \ge n} P(A_m) by Boole's inequality. The right side tends to 00 as n→∞n \to \infty. The lemma needs no independence. The second Borel–Cantelli lemma gives a partial converse for independent events. The lemma holds in each measure space: if ∑nμ(An)<∞\sum_n \mu(A_n) < \infty, then μ(lim sup⁡nAn)=0\mu(\limsup_n A_n) = 0.

History

The lemma is named after Émile Borel and Francesco Paolo Cantelli.

Reading path

To understand this page from the ground up, read these 31 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptRational numbers, object, no evidence levelRational numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionProbability space, definition, no evidence levelProbability spacedefinitionBoole's inequality, statement, no evidence levelBoole's inequalitystatementContinuity of probability measures, statement, no evidence levelContinuity of probabi…statementFirst Borel–Cantelli lemma, statement, no evidence levelFirst Borel–Cantelli …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Set definition
  9. Ring of integers statement
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Union definition
  14. Sample space definition
  15. The rationals form a field statement
  16. Universal quantifier definition
  17. Subset definition
  18. The rationals form an ordered field statement
  19. Power set definition
  20. Universal set definition
  21. Order on the reals definition
  22. Complement definition
  23. Interval definition
  24. Sigma-algebra definition
  25. Event definition
  26. Probability axioms concept
  27. Probability measure definition
  28. Probability space definition
  29. Boole's inequality statement
  30. Continuity of probability measures statement
  31. First Borel–Cantelli lemma statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Borel–Cantelli_lemma, Statement of lemma for probability spaces
About this page
Tag0T6WS7TT77
URIm2://0T6WS7TT77
Aliasesborel-cantelli-lemma
Kindstatement, lemma
WritingW0 stub
MSC60F20
StatusThis copy of the site has no build data. Run m2 status 0T6WS7TT77.
UsesBoole's inequality, Continuity of probability measures, Event

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