= Cumulant expansion
{wiki}
The <cumulant expansion> is the <Taylor expansion> of the <logarithm> of a <moment-generating function>:
$$
K_X(t)=\log\mathbb E e^{tX}
=\sum_{n\geq1}\frac{\kappa_n(X)}{n!}t^n.
$$
If the <moment-generating function> is finite on a <neighbourhood> of zero, the <logarithm> is <analytic function>[analytic] on a sufficiently small <neighbourhood> and this is a convergent local expansion. The first terms are
$$
K_X(t)=t\mathbb EX+\frac{t^2}{2}\operatorname{Var}(X)
+\frac{t^3}{6}\mathbb E[(X-\mathbb EX)^3]+\cdots.
$$
A 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 moment>[exponential integrability] is a separate hypothesis.
For a centered <normal distribution> of <variance> $\sigma^2$, $K_X(t)=\sigma^2t^2/2$, 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 function>[cumulant-generating functions], making <cumulants> additive.
For a fluctuating interaction $V$, expanding a <Boltzmann weight> gives $\log\mathbb E e^{-\beta gV}=-\beta g\mathbb EV+\beta^2g^2\operatorname{Var}(V)/2+\cdots$ whenever the <exponential moment> exists near $g=0$. The <logarithm> thus organizes the expansion into <cumulants> rather than <raw moment>[raw moments].
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