the Edgeworth expansion
/ EDJ-werth /
Berry-Esseen tells you how far a normalized sum is from normal; the Edgeworth expansion tells you what the leading DEVIATIONS look like, term by term, in powers of 1/sqrt(n). It is the systematic refinement of the CLT that corrects the normal approximation for skewness, kurtosis, and higher cumulants of the summands.
Suppose iid X_i have mean 0, variance 1, and enough finite moments. The cdf of S_n = (X_1 + ... + X_n)/sqrt(n) admits the expansion F_n(x) = Phi(x) + phi(x)[ -gamma_1 He_2(x)/(6 sqrt(n)) - (gamma_2 He_3(x)/24 + gamma_1^2 He_5(x)/72)/n + ... ], where Phi and phi are the standard normal cdf and density, He_k are the Hermite polynomials, gamma_1 is the skewness (third cumulant) and gamma_2 the excess kurtosis (fourth cumulant) of X_1. The recipe is pure cf calculus: expand the log of the cf in cumulants, log phi_{S_n}(t) = -t^2/2 + (i t)^3 gamma_1/(6 sqrt(n)) + ..., exponentiate, and invert the Fourier transform term by term; the Hermite polynomials appear because they are the inverse transforms of (it)^k e^(-t^2/2). The first correction is the skewness term of order 1/sqrt(n); for symmetric summands it vanishes and the leading correction is the kurtosis term of order 1/n (which is why symmetric sums look normal faster).
Edgeworth expansions matter in statistics for accurate small-sample p-values, confidence intervals, and as the theoretical justification for the second-order accuracy of the bootstrap. The honest caveat is that the Edgeworth series is an ASYMPTOTIC expansion, not a convergent one: adding more terms improves accuracy for moderate n but the full series typically diverges, and the truncated approximation can even leave the [0,1] range of a valid cdf in the tails. It also requires a lattice/Cramer smoothness condition on X (Cramer's condition limsup |phi(t)| < 1 as |t| -> infinity) for the non-lattice form; lattice variables get a separate, corrected expansion.
For sums of right-skewed variables (gamma_1 > 0, e.g. exponentials), the leading Edgeworth term -gamma_1 He_2(x) phi(x)/(6 sqrt(n)) shifts probability mass to make the sum's distribution skewed compared with the symmetric normal — explaining why a plain normal approximation is too optimistic in the upper tail for small n.
The first correction is the skewness term of order 1/sqrt(n); for symmetric data it vanishes, leaving the 1/n kurtosis term.
The Edgeworth series is asymptotic, not convergent — truncated approximations improve with more terms only up to a point, and can dip below 0 or above 1 in the tails.