Solution
= Solution
Because the variables are <independent random variables>[independent], the moment-generating function of the sum factors. Whenever $|\lambda|<1/\max_i\alpha_i$,
$$
\mathbb E\exp\left(\lambda\sum_{i=1}^nX_i\right)
=\prod_{i=1}^n\mathbb Ee^{\lambda X_i}
\leq\exp\left(\frac{\lambda^2}{2}\sum_{i=1}^n\nu_i\right).
$$
Therefore $\sum_iX_i$ is sub-exponential with parameters
$$
\nu=\sum_{i=1}^n\nu_i,
\qquad
\alpha=\max_{1\leq i\leq n}\alpha_i.
$$