A friendly proof of the Berry-Esseen theorem

TL;DR

This paper offers an accessible, intuitive proof of the Berry-Esseen theorem, extending it to non-i.i.d. variables using Fourier analysis, suitable for graduate teaching.

Contribution

It provides a clear, friendly exposition of the classical Fourier-analytic proof and generalizes the Berry-Esseen theorem beyond i.i.d. cases.

Findings

Provides a simplified proof of the Berry-Esseen theorem

Extends the theorem to non-i.i.d. random variables

Suitable for graduate probability courses

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.