Let $(X_1, \ldots, X_n)$ be a Gaussian vector, and $Z = \sum_{i=1}^n |X_i|$. Since the map $x \mapsto e^{x^2}$, is convex, for any $t>0$ $$ e^{tZ^2} \, = \, e^{t \big(\sum_{i=1}^n |X_i| \big)^2} \, = \, e^{ \big(\frac 1n\sum_{i=1}^n \sqrt {t} \, n|X_i| \big)^2} \, \leq \, \frac 1n \sum_{i=1}^n e^{t n^2 X_i^2} . $$ The law of each $X_i$ has density $\frac {1}{\sqrt {2\pi \sigma_i^2}} \, e^{- (x-m_i)^2/2\sigma_i^2}$, so that there exists $t_0 >0$ such that $$ E \big(e^{t_0Z^2} \big) \, \leq \, \frac 1n \sum_{i=1}^n E \big (e^{t_0 n^2 X_i^2} \big) \, < \, \infty. $$ But is there a way to find the largest $t_0 >0$ such that $E(e^{t_0Z^2}) < \infty$?