Math 2.0

Probability theory and stochastic processes Limit theorems Theorem

Central limit theorem

Builds on Independent and identically distributed random variablesConvergence in distributionNormal distributionLévy's continuity theoremCharacteristic function

Statement

Let X1,X2,…X_1, X_2, \dots be i.i.d. real random variables with mean μ\mu and variance σ2\sigma^2, where 0<σ2<∞0 < \sigma^2 < \infty. Let Xˉn=(X1+⋯+Xn)/n\bar{X}_n = (X_1 + \dots + X_n)/n. Then n (Xˉn−μ)/σ\sqrt{n} \, (\bar{X}_n - \mu) / \sigma converges in distribution to 𝒩(0,1)\mathcal{N}(0, 1). That is, for each x∈ℝx \in \mathbb{R}, lim⁡n→∞P(X1+⋯+Xn−nμσn≤x)=Φ(x)=∫−∞x12πe−u2/2 du.\lim_{n \to \infty} P\left(\frac{X_1 + \dots + X_n - n\mu}{\sigma \sqrt{n}} \le x\right) = \Phi(x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-u^2/2} \, du.

Remarks

A standard proof uses characteristic functions. With Yi=(Xi−μ)/σY_i = (X_i - \mu)/\sigma, φY1(t)=1−t2/2+o(t2)\varphi_{Y_1}(t) = 1 - t^2/2 + o(t^2), so φY1(t/n)n→e−t2/2\varphi_{Y_1}(t/\sqrt{n})^n \to e^{-t^2/2}. Lévy's continuity theorem then gives the convergence in distribution. The Lindeberg and Lyapunov forms remove the condition that the XiX_i have the same law. The Berry–Esseen theorem bounds the rate of convergence when E[∣X1∣3]<∞E[|X_1|^3] < \infty. The finite variance is necessary. Sums of i.i.d. Cauchy variables, scaled by 1/n1/n, stay Cauchy.

Reading path

To understand this page from the ground up, read these 67 pages in order. The foundations come first.

  1. Complex numbers object
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Random experiment concept
  8. Rational numbers object
  9. Real numbers object
  10. Euler's formula statement
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate 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. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Power set definition
  28. Universal set definition
  29. Order on the reals definition
  30. Function definition
  31. Complement definition
  32. Interval definition
  33. Injective function definition
  34. Indicator function definition
  35. Preimage of a set definition
  36. Sigma-algebra definition
  37. Countable set definition
  38. Event definition
  39. Borel sigma-algebra definition
  40. Probability axioms concept
  41. Probability measure definition
  42. Probability distribution definition
  43. Dirac measure definition
  44. Probability space definition
  45. Conditional probability definition
  46. Random variable definition
  47. Cumulative distribution function definition
  48. Discrete random variable definition
  49. Law of a random variable definition
  50. Independence of two events definition
  51. Random variable with a density definition
  52. Convergence in distribution definition
  53. Probability mass function definition
  54. Expected value definition
  55. Identically distributed random variables definition
  56. Mutual independence of events definition
  57. Probability density function definition
  58. Law of the unconscious statistician statement
  59. Independent sigma-algebras definition
  60. Density of the normal distribution definition
  61. Characteristic function definition
  62. Independence of random variables definition
  63. Normal distribution definition
  64. The characteristic function determines the distribution statement
  65. Independent and identically distributed random variables definition
  66. Lévy's continuity theorem statement
  67. Central limit theorem statement

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Central_limit_theorem, Classical CLT
About this page
Tag0EPZC4Z1FC
URIm2://0EPZC4Z1FC
Aliasesclt, lindeberg-levy-central-limit-theorem
Kindstatement, theorem
WritingW1 stated
MSC60F05
StatusThis copy of the site has no build data. Run m2 status 0EPZC4Z1FC.
UsesIndependent and identically distributed random variables, Convergence in distribution, Normal distribution, Lévy's continuity theorem, Characteristic function
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