Edgeworth expansion

Remarks

The proof compares characteristic functions. The characteristic function of $S_n$ is $e^{-t^2/2}\big(1 + \frac{\mu_3}{6\sigma^3\sqrt{n}}(it)^3\big) + o(n^{-1/2})$ for $t$ in ranges of order $\sqrt{n}$, and the correction term is the Fourier transform of the correction in the statement. A smoothing inequality turns closeness of characteristic functions into closeness of distribution functions; the non-lattice hypothesis controls the characteristic function away from $0$.

The central limit theorem says $F_n(x) \to \Phi(x)$; the expansion gives the first correction, of order $1/\sqrt{n}$, which comes from skewness. For a symmetric distribution, $\mu_3 = 0$ and the error of the normal approximation is $o(1/\sqrt{n})$. Under stronger moment conditions and Cramér's condition $\limsup_{\abs{t} \to \infty} \abs{E e^{itX_1}} < 1$, one gets further terms in powers of $1/\sqrt{n}$ with Hermite polynomials and higher cumulants.

The non-lattice hypothesis cannot be dropped. For Bernoulli($1/2$) variables, $F_n$ has jumps of size of order $1/\sqrt{n}$, while the right side is continuous in $x$, so the error cannot be $o(1/\sqrt{n})$ uniformly.

Going deeper

For a student

The Berry–Esseen theorem bounds the error of the normal approximation by $C\, E\abs{X_1 - \mu}^3/(\sigma^3\sqrt{n})$; the Edgeworth expansion identifies the leading term of that error. Expansions of this kind explain why the bootstrap is often more accurate than the normal approximation for studentized statistics. They also give corrected quantiles, through the Cornish–Fisher expansion, and corrected confidence intervals.