Math 2.0

Probability theory and stochastic processes Definition

Convergence in distribution

Builds on Cumulative distribution functionRandom variable

Definition

Let X,X1,X2,…X, X_1, X_2, \dots be real random variables, possibly on different probability spaces, with cumulative distribution functions F,F1,F2,…F, F_1, F_2, \dots. The sequence (Xn)(X_n) converges in distribution to XX if lim⁡n→∞Fn(x)=F(x)\lim_{n \to \infty} F_n(x) = F(x) for each x∈ℝx \in \mathbb{R} at which FF is continuous. One writes Xn→dXX_n \xrightarrow{d} X.

Remarks

By the portmanteau theorem, an equivalent condition is E[g(Xn)]→E[g(X)]E[g(X_n)] \to E[g(X)] for each bounded continuous g ⁣:ℝ→ℝg \colon \mathbb{R} \to \mathbb{R}. This form extends to random variables in metric spaces. The condition depends only on the laws. So it is the weak convergence of the laws of the XnX_n to the law of XX. The restriction to continuity points is necessary. For Xn=1/nX_n = 1/n and X=0X = 0, Fn(0)=0F_n(0) = 0 for all nn, but F(0)=1F(0) = 1. The central limit theorem is a result about convergence in distribution.

Reading path

To understand this page from the ground up, read these 41 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 distribution definition
  38. Probability space definition
  39. Random variable definition
  40. Cumulative distribution function definition
  41. Convergence in distribution definition

Where this leads

22 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Convergence_of_random_variables, Convergence in distribution
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Kinddefinition
WritingW1 stated
MSC60F05
DefinesConvergence in distribution
UsesCumulative distribution function, Random variable

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