Math 2.0

Probability theory and stochastic processes Definition

Almost sure convergence

Builds on Almost surelyRandom variable

Definition

Let X,X1,X2,…X, X_1, X_2, \dots be real random variables on one probability space (Ω,ℱ,P)(\Omega, \mathcal{F}, P). The sequence (Xn)(X_n) converges almost surely to XX if P({ω∈Ω:lim⁡n→∞Xn(ω)=X(ω)})=1.P\left(\left\{\omega \in \Omega : \lim_{n \to \infty} X_n(\omega) = X(\omega)\right\}\right) = 1. One writes Xn→XX_n \to X a.s. or Xn→a.s.XX_n \xrightarrow{\text{a.s.}} X.

Remarks

The set in the definition is an event, because the XnX_n and XX are measurable. An equivalent condition: for each ε>0\varepsilon > 0, P(∣Xn−X∣>ε for infinitely many n)=0P(|X_n - X| > \varepsilon \text{ for infinitely many } n) = 0. The limit is unique up to equality almost surely. The strong law of large numbers is a result about almost sure convergence.

Reading path

To understand this page from the ground up, read these 40 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 numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionPower set, definition, no evidence levelPower setdefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionInterval, definition, no evidence levelIntervaldefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionBorel sigma-algebra, definition, no evidence levelBorel sigma-algebradefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionProbability space, definition, no evidence levelProbability spacedefinitionAlmost surely, definition, no evidence levelAlmost surelydefinitionRandom variable, definition, no evidence levelRandom variabledefinitionAlmost sure convergence, definition, no evidence levelAlmost sure convergen…definition
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. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Preimage of a set definition
  32. Sigma-algebra definition
  33. Event definition
  34. Borel sigma-algebra definition
  35. Probability axioms concept
  36. Probability measure definition
  37. Probability space definition
  38. Almost surely definition
  39. Random variable definition
  40. Almost sure convergence definition

Where this leads

12 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Convergence_of_random_variables, Almost sure convergence
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