= Quadratic expansion of a cumulant-generating function
{title2=$\log M_X(t)=\mu t+\frac12\sigma^2t^2+o(t^2)$}
Suppose the <moment-generating function> $M_X$ is finite on an open <interval> around zero. Its <cumulant-generating function> $K_X=\log M_X$ satisfies $K_X(0)=0$, $K_X'(0)=\mathbb E X=\mu$ and $K_X''(0)=\operatorname{Var}(X)=\sigma^2$. Indeed,
$$
K_X'=\frac{M_X'}{M_X},\qquad K_X''=\frac{M_X''}{M_X}-\left(\frac{M_X'}{M_X}\right)^2,
$$
and $M_X(0)=1$. The <Taylor expansion> gives the displayed formula, with <little o notation> describing the remainder. Finiteness in a neighborhood justifies differentiation: polynomial factors times $e^{tX}$ are dominated on a smaller interval by constants times two slightly larger <exponential moments>.
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