the particular property of the stable distribution, i was thinking of, is "closure under convolution" which is the above marginalisation (i believe?).
infinite divisibility is (yet another) property of gaussians!
infinite divisibility is (yet another) property of gaussians!
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