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A friendly proof of the Berry-Esseen theorem

Roman Vershynin

TL;DR

This note addresses non-asymptotic Berry-Esseen bounds for sums $S_n$ of independent mean-zero variables with $\mathrm{Var}(S_n)=1$. It presents a friendly Fourier-analytic proof centered on smoothing the indicator, Fourier inversion, and a Taylor expansion, culminating in Esseen's smoothing inequality to compare $S_n$ with a standard normal $G$. A key contribution is a general bound $\sup_a |\mathbb{P}(S_n\le a)-\mathbb{P}(G\le a)| \le C\rho^3$ with $\rho^3=\sum_{k=1}^n \mathbb{E}|X_k|^3$, which yields the classical $O(n^{-1/2})$ rate in the i.i.d. unit-variance setting and extends to non-i.i.d. cases. The approach clarifies the role of smoothing and the characteristic function in obtaining uniform, explicit convergence rates in the central limit theorem for dependent or heterogeneous sums, making the proof accessible for graduate coursework.

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.

A friendly proof of the Berry-Esseen theorem

TL;DR

This note addresses non-asymptotic Berry-Esseen bounds for sums of independent mean-zero variables with . It presents a friendly Fourier-analytic proof centered on smoothing the indicator, Fourier inversion, and a Taylor expansion, culminating in Esseen's smoothing inequality to compare with a standard normal . A key contribution is a general bound with , which yields the classical rate in the i.i.d. unit-variance setting and extends to non-i.i.d. cases. The approach clarifies the role of smoothing and the characteristic function in obtaining uniform, explicit convergence rates in the central limit theorem for dependent or heterogeneous sums, making the proof accessible for graduate coursework.

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.
Paper Structure (9 sections, 6 theorems, 69 equations)

This paper contains 9 sections, 6 theorems, 69 equations.

Table of Contents

  1. The proof strategy
  2. Smoothing.
  3. Fourier transform.
  4. Taylor approximation.
  5. A challenge and a fix.
  6. Background on Fourier transform
  7. Smoothing
  8. Characteristic functions
  9. Proof of Theorem \ref{['thm: berry-esseen']}

Key Result

Theorem 1

Let $X_{1}, \dots, X_{n}$ be independent mean-zero random variables whose sum $S_{n} \coloneqq X_{1} + \cdots + X_{n}$ satisfies $\mathop{\mathrm{Var}}\nolimits(S_{n}) = 1$. Let $G$ be a standard normal random variable. Then provided the right-hand side is finite, where $C$ is an absolute constant.

Theorems & Definitions (11)

  • Theorem 1: Berry--Esseen theorem BerryEsseen
  • Lemma 3.1: Smoothing the discrepancy
  • proof
  • Lemma 3.2: Smoothed discrepancy via characteristic functions
  • proof
  • Theorem 3.3: Esseen's smoothing inequality Esseen smoothing
  • Lemma 4.1: The characteristic function of a random variable
  • proof
  • Remark 4.2: No approximation everywhere
  • Lemma 4.3: The characteristic function of a sum
  • ...and 1 more