A friendly proof of the Berry-Esseen theorem

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.

💡 Research Summary

The paper presents a clear and pedagogical proof of the classical Berry‑Esseen theorem using Fourier analysis. The theorem gives a non‑asymptotic bound on the distance between the distribution of a normalized sum of independent (not necessarily identically distributed) random variables and the standard normal law. The author’s goal is to make the Fourier‑analytic proof accessible to graduate students with only basic knowledge of Fourier transforms.

The exposition is organized around four logical steps. First, the discontinuous indicator function 1_{(-∞,a]} is smoothed by convolving it with a Schwartz‑class probability density φ, producing a smooth function f. Lemma 3.1 quantifies the error introduced by this smoothing: the supremum of the distribution‑function difference is bounded by the difference of the smoothed expectations plus a term proportional to the smoothing width ε and the supremum of the density of the comparison variable.

Second, the smoothness of f allows the use of the Fourier inversion formula. By writing E f(S_n) and E f(G) (where G∼N(0,1)) as integrals of the characteristic functions multiplied by the Fourier transform \hat f, the author reduces the problem to bounding an integral of |E e^{itS_n}−e^{-t^2/2}|·|\hat f(t)|. Lemma 3.2 shows that if \hat f is supported on a compact interval (which can be ensured by a suitable choice of φ), the integral can be restricted to |t|≤1/ε and the factor |\hat f(t)| can be bounded by a constant times |t|^{-1}.

Third, the characteristic function of each summand X_k is approximated by a second‑order Taylor expansion. Lemma 4.1 proves that for |t|≤1/ρ_k (where ρ_k^3=E|X_k|^3) we have
E e^{itX_k}=exp(−σ_k^2 t^2/2)+O(ρ_k^3|t|^3).
The proof uses the elementary expansion e^{ix}=1+ix−x^2/2+O(|x|^3) and a logarithmic expansion ln(1+z)=z+O(z^2). The O‑notation hides absolute constants.

Fourth, independence allows the characteristic function of the sum S_n to be written as the product of the individual characteristic functions. Lemma 4.3 combines the previous bounds to obtain, for |t|≤c ρ (ρ=∑_{k=1}^n ρ_k^3), the estimate
|E e^{itS_n}−e^{-t^2/2}| ≤ C ρ |t|^3 e^{-t^2/4}.
When |t| exceeds this range, a trivial bound |E e^{itS_n}|≤1 together with the inequality σ_k≤ρ_k yields the same exponential decay.

Finally, the author applies Esseen’s smoothing inequality (Theorem 3.3) with ε=c/ρ. Because the standard normal density is bounded, the smoothing term contributes C ε, while the integral term is bounded by C ρ after substituting the estimate from Lemma 4.3. This yields the desired Berry‑Esseen bound
sup_{a∈ℝ}|P{S_n≤a}−Φ(a)| ≤ C ρ = C ∑_{k=1}^n E|X_k|^3,
where Φ denotes the standard normal distribution function. The constant C is absolute and does not depend on n or the distributions of the X_k.

The paper also includes a brief review of Fourier‑transform basics (definition, inversion, Plancherel identity, convolution theorem) to make the proof self‑contained. Throughout, the author emphasizes intuition: smoothing removes the discontinuity of the indicator, compact support of \hat φ limits the frequency range that needs to be controlled, and the third absolute moment naturally appears as the measure of non‑Gaussianity. The presentation is deliberately elementary, avoiding advanced tools such as Stein’s method, and is therefore suitable for inclusion in a first‑year graduate probability course.