Second-Order Kurtosis and Skewness Corrections

Why a Second-Order Correction Has Two Parts

The central limit theorem describes the leading shape of a standardized sum: a standard normal density. When the summands are not normal, their skewness produces a correction of order \(n^{-1/2}\). At the next order, \(n^{-1}\), two effects appear: the fourth cumulant of one summand and the square of its third cumulant. The second effect arises because exponentiating the cumulant expansion creates a product of two first-order skewness terms.

This tutorial derives those density terms and calculates them for specified sums. The approximation concerns densities, not cumulative distribution functions. It is most useful as a local correction near the central part of a distribution; it should not be treated as an exact density or automatically trusted far into the tails.

The distinction between \(\kappa_4\) and the fourth standardized moment matters: the latter is \(E[X_i^4]\), while the fourth cumulant subtracts \(3\). For a standard normal variable, \(E[X_i^4]=3\), so \(\kappa_4=0\). We use the probabilists' Hermite polynomials:

Deriving the Density Terms

Let \(\varphi_n(t)=E[e^{itS_n}]\) be the characteristic function of \(S_n\). For a mean-zero, variance-one summand with finite fourth moment, the logarithm of its characteristic function has, near zero, the cumulant expansion

Independence gives \(\varphi_n(t)=\{E[e^{itX_1/\sqrt n}]\}^n\). Substituting \(u=t/\sqrt n\) and multiplying the logarithm by \(n\), for each fixed \(t\) we obtain

To recover the characteristic function rather than its logarithm, set \(a=\kappa_3(it)^3/6\) and \(b=\kappa_4(it)^4/24\). Expanding the exponential through order \(n^{-1}\) gives \(1+a/\sqrt n+(b+a^2/2)/n\). In particular, \(a^2/2=\kappa_3^2(it)^6/72\). Thus the Fourier-side expansion is

The word “formal” is important here. Finite fourth moment establishes the fixed-\(t\) coefficient expansion, but by itself does not justify integrating the remainder over all frequencies to prove a density error bound. Turning Fourier coefficients into a rigorous uniform density approximation requires additional regularity and remainder control. The calculation below identifies the Edgeworth terms; it does not claim those stronger error bounds.

The conversion uses the Fourier inversion convention \(f(x)=(2\pi)^{-1}\int e^{-itx}\varphi(t)\,dt\). With that convention, the inverse transform of \((it)^r e^{-t^2/2}\) is \(H_r(x)\phi(x)\). Indeed, the \(r\)th derivative of \(\phi\) is \((-1)^rH_r\phi\); integration by parts in the Fourier transform shows that multiplication by \((it)^r\) in the integrand corresponds to \((-1)^r\phi^{(r)}\), which is \(H_r\phi\). Applying this identity to powers \(3,4,6\) gives the displayed density terms. \(\square\)

The expansion includes the first-order skewness correction as well as both second-order terms. The fourth-cumulant contribution is proportional to \(H_4\); the squared-skewness contribution is proportional to \(H_6\). Omitting the latter would discard the \(a^2/2\) term created when the logarithmic characteristic function is exponentiated.

Worked Density Calculations

Interpreting the Terms and Avoiding Common Errors

The two \(n^{-1}\) contributions represent distinct features of the summand distribution. If \(\kappa_3=0\), the squared-skewness term disappears, but the fourth-cumulant term can remain, as in the uniform example. If \(\kappa_4=0\) but \(\kappa_3\ne0\), the squared-skewness term remains. If both cumulants vanish, the expansion has no corrections through this order.

A common mistake is to treat “second order” as meaning only the fourth-cumulant term. The correction is obtained by expanding the exponential of the cumulant series, not by reading off its logarithm term by term. Squaring the order-\(n^{-1/2}\) skewness term produces an order-\(n^{-1}\) term, so dropping it gives an incomplete expansion. Another frequent error is to substitute ordinary kurtosis for \(\kappa_4\) without subtracting \(3\).

Finally, an Edgeworth expression is a polynomial times a normal density. The polynomial factor can become negative for some \(x\), even though a genuine density cannot. This signals a limitation of the truncated approximation, especially away from the central region; it is not a reason to reinterpret a negative value as a probability or to alter the derived coefficients.