Cumulant expansion

The cumulant expansion writeswhere ${\kappa }_n$ is the $n$th cumulant. Applied to a fluctuating interaction, it expands the logarithm of the averaged Boltzmann weight in connected expectation values.

An Edgeworth expansion approximates a probability density function by a normal distribution multiplied by Hermite polynomials encoding higher cumulants. For a centered variable of variance ${\sigma }^2$ and third cumulant ${\kappa }_3$,The correction follows from inverse-transforming the cubic term of the cumulant-generating function. It is an asymptotic approximation near the bulk of the distribution; truncation can give a negative result in far tails.

For a standardized iid sum with a regular Edgeworth expansion, the first distribution correction is proportional to $\varphi (t)(t^2-1)$ and is even. Subtracting the two endpoint distribution functions therefore cancels the $n^{-1/2}$ contribution. Fixed symmetric pivot cutoffs can consequently have $O(n^{-1})$ coverage error even though each one-sided approximation has error $O(n^{-1/2})$. This cancellation does not automatically hold for asymmetric or estimated random cutoffs.

The Cornish-Fisher expansion inverts an Edgeworth expansion to approximate quantiles. For a standardized iid sum, let ${\gamma }_1$ be the skewness of one summand and $z_{\alpha }$ the standard normal quantile. The first distribution function correction is $-{\gamma }_1(z^2-1)\varphi (z)/(6\sqrt{n})$. Substituting $y_{\alpha }=z_{\alpha }+c/\sqrt{n}$ and expanding $\Phi$ gives $c={\gamma }_1(z_{\alpha }^2-1)/6$. This is a fixed-quantile asymptotic, with $\alpha$ fixed in $(0,1)$.