Math 2.0

Probability theory and stochastic processes Limit theorems Theorem

Strong law of large numbers

Builds on Independent and identically distributed random variablesAlmost sure convergenceWeak law of large numbersLaw 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 almost surely: P(lim⁡n→∞Xˉn=μ)=1.P\left(\lim_{n \to \infty} \bar{X}_n = \mu\right) = 1.

Remarks

The strong law implies the weak law, because almost sure convergence implies convergence in probability. A short proof exists if E[X14]<∞E[X_1^4] < \infty: then E[(Xˉn−μ)4]≤C/n2E[(\bar{X}_n - \mu)^4] \le C/n^2, and the first Borel–Cantelli lemma applies. Etemadi proved that the result holds when the XiX_i are only pairwise independent and identically distributed. Conversely, if E[∣X1∣]=∞E[|X_1|] = \infty, then almost surely Xˉn\bar{X}_n does not converge to a finite limit.

Reading path

To understand this page from the ground up, read these 78 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. Almost surely definition
  47. Conditional probability definition
  48. Monotonicity of probability statement
  49. Random variable definition
  50. Independence of two events definition
  51. Almost sure convergence definition
  52. Discrete random variable definition
  53. Law of a random variable definition
  54. Random variable with a density definition
  55. Convergence in probability definition
  56. Mutual independence of events definition
  57. Independence of two random variables definition
  58. Joint probability distribution definition
  59. Expected value definition
  60. Identically distributed random variables definition
  61. Independent sigma-algebras definition
  62. Marginal distribution definition
  63. Markov's inequality statement
  64. Linearity of expectation statement
  65. Variance definition
  66. Independence of random variables definition
  67. Independence means the joint law is the product of the marginals statement
  68. Covariance definition
  69. Standard deviation definition
  70. Expectation of a product of independent random variables statement
  71. Independent and identically distributed random variables definition
  72. Chebyshev's inequality statement
  73. Variance of a sum statement
  74. Independent random variables are uncorrelated statement
  75. Law of large numbers concept
  76. Variance of a sum of independent random variables statement
  77. Weak law of large numbers statement
  78. Strong law of large numbers statement

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_large_numbers, Strong law
About this page
Tag0DFWHQ839T
URIm2://0DFWHQ839T
Aliaseskolmogorov-strong-law
Kindstatement, theorem
WritingW1 stated
MSC60F15
StatusThis copy of the site has no build data. Run m2 status 0DFWHQ839T.
UsesIndependent and identically distributed random variables, Almost sure convergence, Weak law of large numbers, Law of large numbers

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