Solution
= Solution
Independent <Poisson process>[Poisson processes] have stationary independent increments, so their weighted sum does too and is stochastically continuous. Moreover,
$$
\mathbb E e^{iuX_t}
=\prod_{k=1}^n
\exp\{t\lambda_k(e^{iua_k}-1)\}
=e^{t\psi(u)},
$$
where
$$
\psi(u)=\sum_{k=1}^n\lambda_k(e^{iua_k}-1).
$$