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.

翻译：我们获得了欧几里得球面上概率分布参数族（如冯·米塞斯-费希尔分布、沃森分布和角高斯分布）的离散混合表示。除若干特殊结果外，我们基于特殊面积谐波的密度展开，提出了一种处理各向同性分布族的通用方法。我们探讨了该表示与球面随机过程（特别是随机游走）的联系、从球面扩散导出的离散混合表示，以及利用马尔可夫表示作为混合基来获得球面分布族表示的方法。