The Central Limit Theorem

Abstract

The law of large numbers states that the arithmetic mean of independent, identically distributed random variables converges to the expected value. One interpretation of the central limit theorem is as a (distributional) rate result. Technically, let X, X₁, X₂, . . . be independent, identically distributed random variables with mean μ. The weak and strong laws of large numbers state that \(\frac{1} {n}\sum\nolimits_{k = 1}^n {X_k \to \mu } \) in probability and almost surely, respectively, as n→∞. A distributional rate result deals with the question of how one should properly “blow up” the difference \( \frac{1} {n}\sum\nolimits_{k = 1}^n {X_k - \mu } \) in order for the limit to have a non-trivial limit as n tends to infinity. The corresponding theorem was first stated by Laplace. The first general version with a rigorous proof is due to Lyapounov [178, 179].