the Berry-Esseen theorem

The classical CLT says the law of normalized sums converges to the normal but is silent on HOW FAST. The Berry-Esseen theorem supplies the missing rate: it bounds, uniformly across the real line, the distance between the exact distribution function of a normalized sum and the standard normal cdf. It is the sharpest first-order quantitative form of the CLT.

Let X_1, ..., X_n be iid with mean 0, variance sigma^2 > 0, and finite third absolute moment rho = E[|X_1|^3]. Let F_n be the cdf of S_n = (X_1 + ... + X_n)/(sigma sqrt(n)) and Phi the standard normal cdf. Then sup_x |F_n(x) - Phi(x)| <= C rho/(sigma^3 sqrt(n)), where C is a universal constant (numerically C < 0.5, with the best known constant around 0.4690). The rate is order 1/sqrt(n), and the dimensionless quantity rho/sigma^3 (a scale-free measure of skewness/heaviness) sets the constant. The proof is a cf argument: Esseen's smoothing inequality bounds the sup-distance of cdfs by an integral of |phi_{S_n}(t) - e^(-t^2/2)|/|t| over a bounded frequency range, and the Taylor expansion of the cf near 0 (using the third moment) controls that integral.

Berry-Esseen matters wherever you need a guarantee, not just an asymptotic: statistical confidence intervals, Monte Carlo error bars, and any setting where n is finite. Two honest caveats. First, the 1/sqrt(n) rate is in general OPTIMAL and cannot be improved by this uniform sup-norm metric — symmetric distributions can converge faster pointwise, but the worst case is 1/sqrt(n) (the binomial near a lattice point is the canonical slow example). Second, the third absolute moment is required; with only a finite variance the CLT still holds (Lindeberg) but you get NO rate from Berry-Esseen, and heavy-tailed laws with infinite third moment can converge arbitrarily slowly.