Math 2.0

Probability theory and stochastic processes Definition

Convergence in mean

Builds on Expected valueRandom variable

Definition

Let r≥1r \ge 1 be real. Let X,X1,X2,…X, X_1, X_2, \dots be real random variables on one probability space with E[∣X∣r]<∞E[|X|^r] < \infty and E[∣Xn∣r]<∞E[|X_n|^r] < \infty for each nn. The sequence (Xn)(X_n) converges in rr-th mean, or in LrL^r, to XX if lim⁡n→∞E[∣Xn−X∣r]=0.\lim_{n \to \infty} E\left[|X_n - X|^r\right] = 0. For r=1r = 1 one says convergence in mean, and for r=2r = 2 convergence in mean square.

Remarks

By Markov's inequality, P(∣Xn−X∣>ε)≤E[∣Xn−X∣r]/εrP(|X_n - X| > \varepsilon) \le E[|X_n - X|^r] / \varepsilon^r. So convergence in rr-th mean implies convergence in probability. For s≥r≥1s \ge r \ge 1, convergence in ss-th mean implies convergence in rr-th mean. Convergence in mean does not imply almost sure convergence, and almost sure convergence does not imply convergence in mean. Example: on [0,1][0, 1] with the uniform measure, Xn=n 1[0,1/n]X_n = n \, 1_{[0, 1/n]} converges to 00 almost surely but E[Xn]=1E[X_n] = 1 for all nn.

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. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Countable set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability space definition
  40. Random variable definition
  41. Discrete random variable definition
  42. Random variable with a density definition
  43. Expected value definition
  44. Convergence in mean definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Convergence_of_random_variables, Convergence in mean
About this page
Tag0TTA57SEJG
URIm2://0TTA57SEJG
Kinddefinition
WritingW1 stated
MSC60F25
DefinesConvergence in mean
UsesExpected value, Random variable

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by