 Cumulant expansion

The cumulant expansion is the Taylor expansion of the logarithm of a moment-generating function:If the moment-generating function is finite on a neighbourhood of zero, the logarithm is analytic on a sufficiently small neighbourhood and this is a convergent local expansion. The first terms areA formal series of moments is not automatically a convergent representation of a moment-generating function. Finitely many moments can instead justify finite-order expansions of the characteristic function and its logarithm near zero; exponential integrability is a separate hypothesis.

For a centered normal distribution of variance ${\sigma }^2$, $K_X(t)={\sigma }^2{t}^2/2$, so all cumulants beyond order two vanish. For a Poisson distribution of mean $\lambda$, $K_X(t)=\lambda (e^t-1)$, so every cumulant equals $\lambda$. independent sums add their cumulant-generating functions, making cumulants additive.

For a fluctuating interaction $V$, expanding a Boltzmann weight gives $\log \mathbb{E}{e}^{-\beta gV}=-\beta g\mathbb{E}V+{\beta }^2{g}^2\mathrm{Var}(V)/2+\cdots$ whenever the exponential moment exists near $g=0$. The logarithm thus organizes the expansion into cumulants rather than raw moments.