[mturmon]

Nope. Closure under convolution is the same as closure under summation of the associated random variable, which is the defining property of stable distributions. This is explained in the first paragraphs of the wikipedia page you linked to ;-)

Closure under marginalization is something else.

It so happens that the functional form of the gaussian satisfies both, but the two properties are not at all the same.

  P1: X, Y gaussian => Z = X+Y gaussian
  P2: X, Y gaussian => X | Y gaussian