Solution
= Solution
Independence makes cumulant-generating functions additive:
$$
\psi_Z(\lambda)=\sum_{i=1}^n\psi_{X_i}(\lambda).
$$
Set
$$
\nu=\sum_{i=1}^n\nu_i,
\qquad
c=\max_{1\leq i\leq n}c_i.
$$
For $0<\lambda<1/c$, every denominator is positive and $1-c_i\lambda\geq1-c\lambda$, so
$$
\psi_Z(\lambda)
\leq\sum_i\frac{\nu_i\lambda^2}{2(1-c_i\lambda)}
\leq\frac{\nu\lambda^2}{2(1-c\lambda)}.
$$
Also $\mathbb EZ=0$, and hence $Z\in\Gamma_+(\nu,c)$.