Math 2.0

Probability theory and stochastic processes Limit theorems Theorem

Weak law of large numbers

Builds on Independent and identically distributed random variablesConvergence in probabilityChebyshev's inequalityVariance of a sum of independent random variablesLaw of large numbers

Statement

Let X1,X2,…X_1, X_2, \dots be i.i.d. real random variables with E[∣X1∣]<∞E[|X_1|] < \infty and μ=E[X1]\mu = E[X_1]. Let Xˉn=(X1+⋯+Xn)/n\bar{X}_n = (X_1 + \dots + X_n)/n. Then Xˉn→μ\bar{X}_n \to \mu in probability: for each ε>0\varepsilon > 0, lim⁡n→∞P(∣Xˉn−μ∣>ε)=0.\lim_{n \to \infty} P\left(\left|\bar{X}_n - \mu\right| > \varepsilon\right) = 0.

Remarks

If σ2=Var⁡(X1)<∞\sigma^2 = \operatorname{Var}(X_1) < \infty, there is a short proof. By the variance of a sum, Var⁡(Xˉn)=σ2/n\operatorname{Var}(\bar{X}_n) = \sigma^2/n. Then Chebyshev's inequality gives P(∣Xˉn−μ∣>ε)≤σ2/(nε2)→0P(|\bar{X}_n - \mu| > \varepsilon) \le \sigma^2 / (n \varepsilon^2) \to 0. The general case uses truncation, or characteristic functions: φXˉn(t)=φX1(t/n)n→eiμt\varphi_{\bar{X}_n}(t) = \varphi_{X_1}(t/n)^n \to e^{i\mu t}. The finite-variance proof needs only that the XiX_i are pairwise uncorrelated with the same mean and bounded variances.

History

Jacob Bernoulli proved the first version, for Bernoulli trials. It appeared after his death in Ars Conjectandi in 1713.

Reading path

To understand this page from the ground up, read these 75 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 bounded above definition
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Supremum definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Completeness axiom statement
  23. Cartesian product definition
  24. Subset definition
  25. The rationals form an ordered field statement
  26. Uniqueness quantifier definition
  27. Existence of square roots of nonnegative reals statement
  28. Binary relation definition
  29. Universal set definition
  30. Order on the reals definition
  31. Power set definition
  32. Square root definition
  33. Complement definition
  34. Function definition
  35. Interval definition
  36. Injective function definition
  37. Preimage of a set definition
  38. Sigma-algebra definition
  39. Countable set definition
  40. Event definition
  41. Borel sigma-algebra definition
  42. Probability axioms concept
  43. Probability measure definition
  44. Probability distribution definition
  45. Probability space definition
  46. Conditional probability definition
  47. Monotonicity of probability statement
  48. Random variable definition
  49. Independence of two events definition
  50. Discrete random variable definition
  51. Law of a random variable definition
  52. Convergence in probability definition
  53. Random variable with a density definition
  54. Mutual independence of events definition
  55. Independence of two random variables definition
  56. Joint probability distribution definition
  57. Expected value definition
  58. Identically distributed random variables definition
  59. Independent sigma-algebras definition
  60. Marginal distribution definition
  61. Markov's inequality statement
  62. Variance definition
  63. Linearity of expectation statement
  64. Independence of random variables definition
  65. Independence means the joint law is the product of the marginals statement
  66. Covariance definition
  67. Standard deviation definition
  68. Expectation of a product of independent random variables statement
  69. Independent and identically distributed random variables definition
  70. Chebyshev's inequality statement
  71. Variance of a sum statement
  72. Independent random variables are uncorrelated statement
  73. Law of large numbers concept
  74. Variance of a sum of independent random variables statement
  75. Weak law of large numbers statement

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_large_numbers, Weak law
About this page
Tag0BFDHF2H7N
URIm2://0BFDHF2H7N
Aliaseskhinchin-law, bernoulli-law-of-large-numbers
Kindstatement, theorem
WritingW0 stub
MSC60F05
StatusThis copy of the site has no build data. Run m2 status 0BFDHF2H7N.
UsesIndependent and identically distributed random variables, Convergence in probability, Chebyshev's inequality, Variance of a sum of independent random variables, Law of large numbers

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