Math 2.0

Probability theory and stochastic processes Independence Theorem

Kolmogorov's zero–one law

Builds on Independence of random variablesIndependent sigma-algebrasSigma-algebra

Statement

Let X1,X2,…X_1, X_2, \dots be independent random variables on a probability space (Ω,ℱ,P)(\Omega, \mathcal{F}, P). Let 𝒯=⋂n=1∞σ(Xn,Xn+1,… )\mathcal{T} = \bigcap_{n=1}^{\infty} \sigma(X_n, X_{n+1}, \dots) be the tail σ-algebra, where σ(Xn,Xn+1,… )\sigma(X_n, X_{n+1}, \dots) is the smallest σ-algebra that makes XmX_m measurable for all m≥nm \ge n. Then P(A)∈{0,1}P(A) \in \{0, 1\} for each A∈𝒯A \in \mathcal{T}.

Remarks

An event is in 𝒯\mathcal{T} if changing finitely many of the XnX_n cannot change whether it occurs. Examples of tail events: "the series ∑nXn\sum_n X_n converges" and "lim sup⁡n(X1+⋯+Xn)/n≤c\limsup_n (X_1 + \dots + X_n)/n \le c" for a real cc. Proof idea: 𝒯\mathcal{T} is independent of σ(X1,…,Xn)\sigma(X_1, \dots, X_n) for each nn, hence of σ(X1,X2,… )\sigma(X_1, X_2, \dots), which contains 𝒯\mathcal{T}. So each A∈𝒯A \in \mathcal{T} is independent of itself, and P(A)=P(A)2P(A) = P(A)^2. The theorem says that a tail event is almost sure or almost impossible. It does not say which of the two cases holds.

History

The theorem is named after Andrey Kolmogorov.

Reading path

To understand this page from the ground up, read these 44 pages in order. The foundations come first.

  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. Conditional probability definition
  39. Random variable definition
  40. Independence of two events definition
  41. Mutual independence of events definition
  42. Independent sigma-algebras definition
  43. Independence of random variables definition
  44. Kolmogorov's zero–one law statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Kolmogorov's_zero–one_law, Formulation
About this page
Tag073Q8Z94KG
URIm2://073Q8Z94KG
Aliaseskolmogorov-0-1-law, kolmogorov-zero-one-law
Kindstatement, theorem
WritingW0 stub
MSC60F20
StatusThis copy of the site has no build data. Run m2 status 073Q8Z94KG.
UsesIndependence of random variables, Independent sigma-algebras, Sigma-algebra
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