i wonder if you are thinking of another monoid property of a gaussian?

suppose we have:

   x ~ N(0,1)
   y|x ~ N(x, 1)

then we have:

   y ~ N(0, 2)

i.e., gaussians are closed under marginalization.

however, i believe gaussians are not the only distribution with this property either: i think this property corresponds to the stable family of distributions: https://en.wikipedia.org/wiki/Stable_distributions