= Cumulant composition for a random sum
{title2=$\kappa_S(t)=\kappa_N(\kappa_X(t))$}
For a <random sum of independent claims>, conditioning on the count gives $M_S(t)=\mathbb E[M_X(t)^N]$ and therefore the displayed <cumulant-generating function>. Write $a_j=\kappa_{N,j}$ and $b_j=\kappa_{X,j}$. The chain rule gives $\kappa_{S,1}=a_1b_1$, $\kappa_{S,2}=a_1b_2+a_2b_1^2$, and $\kappa_{S,3}=a_1b_3+3a_2b_1b_2+a_3b_1^3$. In raw severity <moments> $m_j=\mathbb EX^j$, the third <cumulant> is $a_1m_3+3(a_2-a_1)m_1m_2+(a_3-3a_2+2a_1)m_1^3$. Positive exponential moments justify two-sided differentiation near zero; finite ordinary moments alone suffice for the corresponding left derivatives when claims are positive.
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