  We obtain discrete mixture representations for parametric families of
probability distributions on Euclidean spheres, such as the von Mises--Fisher,
the Watson and the angular Gaussian families. In addition to several special
results we present a general approach to isotropic distribution families that
is based on density expansions in terms of special surface harmonics. We
discuss the connections to stochastic processes on spheres, in particular
random walks, discrete mixture representations derived from spherical
diffusions, and the use of Markov representations for the mixing base to obtain
representations for families of spherical distributions.
