A friendly proof of the Berry-Esseen theorem

Abstract

A gem of classical probability, the Berry-Esseen theorem provides a non-asymptotic form of the central limit theorem. This note gives a friendly and intuitive exposition of the classical Fourier-analytic proof of Esseen's smoothing inequality and, as a consequence, a general Berry-Esseen theorem for non-i.i.d random variables. The exposition is suitable for use in a basic graduate course in probability.