Will normal random variables form a normal random vector? If $X_1, ..., X_n$ are normal random variables, will $[X_1, ..., X_n]$ be a normal random vector?
I know if $X_1, ..., X_n$  are further independent, then $[X_1, ..., X_n]$ is a normal random vector. Can there be more relaxed condition on $X_1, ..., X_n$ than being independent?
Thanks in advance!

$(X_1,\ldots,X_n)$ is a normal vector if and only if any linear combination $\sum\nolimits_{i = 1}^n {a_i X_i }$, $a_i \in \mathbb{R}$, is a univariate normal variable (here, constants are also regarded as normal variables).
Concerning the first question, let $X$ be an arbitrary Gaussian process indexed by $T$ (for example a Brownian motion or bridge, where $T=[0,\infty)$ or $T=[0,1]$, respectively). Then, by definition, $(X_{t_1},\ldots,X_{t_n})$ is a normal vector, for any $n \geq 1$ and $t_1,\ldots,t_n \in T$, but the components are (in general) dependent random variables.
