Association and infinite divisibility for the Wishart distribution and its diagonal marginals

We give an example of a central Wishart matrix W with one degree of freedom and scale matrix of rank 2 such that the diagonal entries of W are not associated. This allows us to conclude that no central Wishart matrix with one degree of freedom and scale matrix of rank greater than 1 is associated. We also employ the connection between association and infinite divisibility to show that, despite former evidence to the contrary, there exist Gaussian vectors such that the vector of squares is not infinitely divisible. Similarly, we obtain another proof of Lévy's result that no central Wishart matrix with scale matrix of rank greater than 1 is infinitely divisible.