Is a random vector multivariate normal if and only if every linear combination of its coordinates is normal?

Question

The first sentence of this Wikipedia article (https://en.wikipedia.org/wiki/Gaussian_process) seems to imply that a vector of random variables is multivariate normal if (and only if) every linear combination of its coordinates is normal. (I know the only if part is true) Is the if part true? How can one prove it? Thanks!

Answer 1

Suppose all linear combinations of $X_1,X_2,\cdots,X_n$ are normal. Let $t_1,t_2,\cdots,t_n$ be real numbers and $Y=\sum_{k=1}^{n} t_kX_k$. Then $Ee^{isY}=Ee^{is\sum_{k=1}^{n} t_kX_k}=e^{is\mu} e^{-s^{2} \sigma^{2}/2}$ where $\mu =E\sum_{k=1}^{n} t_kX_k$ and $\sigma ^{2}$ is the variance of $\sum_{k=1}^{n} t_kX_k$. If you compute this you get $e^{is\mu}e^{-x^{T}\Sigma x}$ where $x$ is the vector $(t_1,t_2,\cdots,t_n)$ and $\Sigma $ is a non -negative definite matrix. This is the general form of multivariate normal characteristic function. [You can take $s=1$ in this argument].