As pointed out in this question, linear combinations of normal distributed variables are normal distributed:
If $X$ is distributed normally as $N_p(\mu, \Sigma)$ then any linear combination of variables $a'X = a_1X_1 + a_2X_2 + \cdots + a_pX_p$ is distributed as $N(a'\mu, a'\Sigma a)$. Also, if $a'X$ is distributed as $N(a'\mu, a'\Sigma a)$ for every $a$, then $X$ must be $N_p(\mu, \Sigma)$.
Can this result be generalized to broader classes of probability distributions as the exponential family?