Random-sum transform identity ($M_S(t)=G_N(M_X(t))$)

For independent and identically distributed random variables $X_j$ independent of a nonnegative integer count $N$, set $S={\sum }_{j=1}^N{X}_j$, including $S=0$ when $N=0$. Conditioning on the count gives $\mathbb{E}[e^{tS}\mid N=k]=M_X(t{)}^k$, hence the displayed composition of the moment-generating function with the probability generating function. For positive summands the identity holds for all $t\le 0$; positive $t$ requires convergence of the composed series. This permits Laplace transforms of nonnegative random variables to identify the law even when positive exponential moments do not exist.

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 ${\kappa }_{S,1}=a_1{b}_1$, ${\kappa }_{S,2}=a_1{b}_2+a_2{b}_1^2$, and ${\kappa }_{S,3}=a_1{b}_3+3a_2{b}_1{b}_2+a_3{b}_1^3$. In raw severity moments $m_j=\mathbb{E}{X}^j$, the third cumulant is $a_1{m}_3+3(a_2-a_1)m_1{m}_2+(a_3-3a_2+2a_1)m_1^3$. Positive exponential moments justify two-sided differentiation near zero; finite ordinary moments alone suffice for the corresponding left derivatives when claims are positive.