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An Edgeworth expansion refines the central limit theorem by adding corrections determined by higher cumulants. Let $X_1,X_2,\dots$ be independent and identically distributed with mean zero, variance ${\sigma }^2>0$ and skewness ${\gamma }_1=\mathbb{E}{X}_1^3/{\sigma }^3$. For the standardized sum $Z_n=(X_1+\cdots +X_n)/(\sigma \sqrt{n})$, the first density correction isHere $\varphi$ is the standard normal density. This is an asymptotic statement for a sequence, not a universal density formula for one arbitrary random variable. Sufficient smoothness and moment conditions are needed; one strong sufficient setting is that the common probability density function is a non-degenerate Schwartz function. In that setting the displayed remainder is valid for each fixed $z$. Lattice-valued variables need additional corrections rather than this density formula.^[ref]

To identify the coefficient, expand the log characteristic function of the standardized sum:for fixed $t$. Exponentiation and Fourier inversion turn $(it{)}^3{e}^{-t^2/2}$ into the Hermite polynomial ${\mathrm{He}}_3(z)=z^3-3z$ times $\varphi (z)$. The smoothness assumptions justify inversion beyond this formal calculation. Higher-order terms use further cumulants and Hermite polynomials.

The approximation improves central asymptotics, but it need not be a nonnegative normalized density after truncation. In particular the cubic correction can become negative in a far tail, where a fixed-point expansion does not provide a small relative error. Inverting a distribution-function Edgeworth expansion produces the Cornish-Fisher expansion for fixed quantiles.

Peter Gavin Hall, also cited as Peter Hall, developed theory for the bootstrap, nonparametric statistics and approximation of sampling distributions. His professional society's memorial explicitly verifies the fuller identity and these research areas.^[ref]

His The Bootstrap and Edgeworth Expansion by Peter Hall explains how Edgeworth expansions refine a normal approximation using higher cumulants. Such expansions help analyze the error of a bootstrap approximation, including whether a transformed or studentized statistic achieves a higher order of accuracy. Smoothness and sampling assumptions matter: the presence of a formal expansion alone does not establish a uniformly accurate approximation.^[ref]