Equivalent definitions of multivariate normal

Question

Suppose our definition for $(X,Y)$ having the bivariate normal distribution with mean $\mu$ and covariance matrix $\Sigma$ is that any linear combination of $X$ and $Y$ is also normally distributed. Is there any immediate way to see that $(X,Y)$ has the well-known joint pdf? Getting from the joint pdf to the fact that any linear combination is normally distributed is pretty easy, but I can't find a nice way to derive the joint pdf from scratch given the linear combinations definition.

Answer 1

This is done using two dimensional characteristic functions.Assuming, for simplicity, that the means are $0$, $Ee^{i(tX+sY)}=e^{-Var (tX+sY)/2}$ and $E(tX+sY)^{2}$ is a quadratic form in $t$ and $s$. Comparing this with the characteristic function of the joint normal distribution with variance-covariance matrix $\Sigma $ we can conclude that $(X,Y)$ is jointly normal.