Suppose $X\_i, \; i=1,2,3...,n$ are each Gaussian, then it is not in general true that the set is jointly Gaussian (a multivariate Gaussian).


Does a similar statement hold if the variates are *pairwise* Gaussian? i.e. if we have that $X\_i, X\_j$ are a bivariate Gaussian for all $i, j$, then it is not in general true that the set is jointly Gaussian. (And I suppose any dimensional variant of this? k-wise Gaussian => n-wise Gaussian)


If the statement holds, what is a simple low-dimensional example where you have a pairwise Gaussian system where this does not hold? Maybe this is in Counterexamples in Probability ... but I don't have this available right now. 



Pairwise Gaussian vs Jointly Gaussian (k-wise Gaussian vs n-wise Gaussian)
Let $X\_i$, $i=1\ldots3$, be iid standard normal random variables. Let $Y = |X\_3|$ if 
$X\_1 X\_2 > 0$ and $-|X\_3|$ if $X\_1 X\_2 \le 0$. Then $X\_1, X\_2, Y$ are normal and pairwise 
independent, but they are not jointly normal since $X\_1 X\_2 Y \ge 0$.


