Pairwise Gaussian vs Jointly Gaussian (k-wise Gaussian vs n-wise Gaussian)

Question

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.

Answer 1

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 $X_1 X_2 Y \ge 0$.

Answer 2

Another, more general class of examples follows through the construction in "Families of m-Variate distributrions with given margins and m(m-1)/2 Bivariate dependence paramters" by H. Joe 96: The uniform and bivariate margins can be Gaussian, the higher level copulas can be arbitrary.