Math 2.0

Probability theory and stochastic processes Limit theorems Lemma

Second Borel–Cantelli lemma

Builds on First Borel–Cantelli lemmaMutual independence of eventsProbability of the complement

Statement

Let A1,A2,…A_1, A_2, \dots be mutually independent 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)=1.P\left(\limsup_{n \to \infty} A_n\right) = 1. So almost surely, infinitely many of the events AnA_n occur.

Remarks

Proof: the complement of lim sup⁡nAn\limsup_n A_n is ⋃n⋂m≥nAmc\bigcup_n \bigcap_{m \ge n} A_m^c. By independence and 1−x≤e−x1 - x \le e^{-x}, P(⋂m=nNAmc)=∏m=nN(1−P(Am))≤exp⁡(−∑m=nNP(Am))P\left(\bigcap_{m = n}^{N} A_m^c\right) = \prod_{m=n}^{N} (1 - P(A_m)) \le \exp\left(-\sum_{m=n}^{N} P(A_m)\right). The right side tends to 00 as N→∞N \to \infty. So each ⋂m≥nAmc\bigcap_{m \ge n} A_m^c has probability 00, and so does their union. Independence is necessary. For An=AA_n = A with 0<P(A)<10 < P(A) < 1 for all nn, the sum diverges but P(lim sup⁡nAn)=P(A)<1P(\limsup_n A_n) = P(A) < 1. With the first lemma, it follows that for independent events, P(lim sup⁡nAn)P(\limsup_n A_n) is 00 or 11.

Reading path

To understand this page from the ground up, read these 36 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 inequalitystatementConditional probability, definition, no evidence levelConditional probabili…definitionContinuity of probability measures, statement, no evidence levelContinuity of probabi…statementProbability of the complement, statement, no evidence levelProbability of the co…statementIndependence of two events, definition, no evidence levelIndependence of two e…definitionFirst Borel–Cantelli lemma, statement, no evidence levelFirst Borel–Cantelli …statementMutual independence of events, definition, no evidence levelMutual independence o…definitionSecond Borel–Cantelli lemma, statement, no evidence levelSecond 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. Conditional probability definition
  31. Continuity of probability measures statement
  32. Probability of the complement statement
  33. Independence of two events definition
  34. First Borel–Cantelli lemma statement
  35. Mutual independence of events definition
  36. Second Borel–Cantelli lemma statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Borel–Cantelli_lemma, Converse result
About this page
Tag0NWXRAMEE7
URIm2://0NWXRAMEE7
Aliasesconverse-borel-cantelli-lemma
Kindstatement, lemma
WritingW0 stub
MSC60F20
StatusThis copy of the site has no build data. Run m2 status 0NWXRAMEE7.
UsesFirst Borel–Cantelli lemma, Mutual independence of events, Probability of the complement

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