The normal distribution has the property that if X and Y are i.i.d. and are both normally distributed, then X + Y will also be normally distributed. My question is about directional statistics. Consider probability distributions on the unit circle, where all points can be parametrized by angle. The wrapped normal distribution also has the property that if X and Y are angles that are i.i.d., their sum will be distributed according to the wrapped normal distribution. Note that the von Mises distribution does not have this property, but since the von Mises distribution is fairly close in shape to the wrapped normal distribution (and is often used in place of it), some confuse the two. Here is my question: what is the extension of this idea to the sphere? Is it possible to come up with a distribution P, and a parametrization of points on the sphere, in such a way that if X and Y are the parametrizations of two random variables that are i.i.d. and distributed according to P, X+Y will also be distributed according to P?