The Central Limit Theorem

Abstract

The Central Limit Theorem is one of the most impressive achievements of probability theory. Prom a simple description requiring minimal hypotheses, we are able to deduce precise results. The Central Limit Theorem thus serves as the basis for much of Statistical Theory. The idea is simple: let X₁,...,X _j ,... be a sequence of i.i.d. random variables with finite variance. Let S _n = Σ ⁿ_j=1 X _j . Then for n large, L(S _n ) ≈ N(nμ, nσ²), where E{X _j } = μ and σ² = Var(X_j) (all j). The key observation is that absolutely nothing (except a finite variance) is assumed about the distribution of the random variables (X_j)_{j ≥ 1}. Therefore, if one can assume that a random variable in question is the sum of many i.i.d. random variables with finite variances, that one can infer that the random variable’s distribution is approximately Gaussian. Next one can use data and do Statistical Tests to estimate μ and σ², and then one knows essentially everything!