Infinite divisibility of joint distribution assuming that of the marginals

Question

Assume $F$ is a multivariate dsitribution on $\mathbb R^n$ such that for all $i=1,\ldots,n$ its marginals $F_i$ are infinitely-divisible on $\mathbb R$. Is $F$ necessarily infinitely-divisible?

Answer 1

The answer is no. Here is an easy counterexample in $\mathbb R^2$:

Let $X\sim \mathcal N(0,1)$ and let $B$ be a random variable independent of $X$ with $\mathbb P[B=+1]=\mathbb P[B=-1]=\frac12$. Define $Y=BX$, then the vector $(X,Y)$ has normal (i.e. infinitely divisible) marginals, however $(X,Y)$ is not infinitely divisible.

To see this, note that $(X,Y)$ is concentrated on the diagonals $$D:=\{(x,y)\in\mathbb R^2: |x|=|y|\}.$$ I claim that from this it immediately follows that $(X,Y)$ cannot be infinitely divisible: Note that if we add two independent vectors $Z_1,Z_2$, the support of $Z_1+Z_2$ will be equal to the Minkowski sum of the support of $Z_1$ and the support of $Z_2$. Then in order to see that $(X,Y)$ is not infinitely divisible, it is enough to see that the set $D$ cannot be written as a Minkowski sum $A+A=D$, for any set $A$.