Deriving the central limit theorem from the de Moivre–Laplace theorem

Abstract

The de Moivre–Laplace theorem is a special case of the central limit theorem for Bernoulli random variables, and can be proved by direct computation. We deduce the central limit theorem for any random variable with finite variance from the de Moivre–Laplace theorem. Our proof does not use advanced notions such as characteristic functions, the Brownian motion, or stopping times.