Given a random vector $(X\_1,X\_2)$. If $aX\_1 + bX\_2$ is Gaussian for all pairs $a,b$, then $(X\_1,X\_2)$ is jointly normal. More generally, is the following statement true?
If $aX\_1 + bX\_2$ has the same distribution as $aY\_1 + bY\_2$, for all $a,b$, then $(X\_1,X\_2)$ has the same distribution as $(Y\_1,Y\_2)$. I know this is true if $(Y\_1,Y\_2)$ is the pullback under some diffeomorphism of a jointly normal vector. But is it true in general?


Also if I have total variation bound on the difference between $aX\_1 + bX\_2$ and $aY\_1 + bY\_2$ for all $a,b$, say above by $\epsilon$, does that give any total variation bound between the joint distributions?



when does inner product with fixed vectors determine joint distribution?
The first question has an affirmative answer. If two 2-dimensional distributions have the same (2-dim) characteristic functions they coincide. The characteristic function of the 2-dimensional distribution of (X,Y) is determined only by the distributions of aX+bY for all a and b. 
The proof of that fact (Fourier inversion formula) may help to solve the second question as well.


