A Generalization of Weyl's Integration Theorem and Its Meaning for Stochastic Simulations

Due to Weyl's integration theorem the Haar probability measure and, further, a whole class of probability measures on a compact connected Lie group G can be represented as image measures of product measures of a specific type. It will be shown that this result holds even for a larger class of probability measures on G. As a consequence, a simulation of any distribution which is contained in this (larger) class can be decomposed into two simulation problems of smaller size which can be treated independently. This aspect will be investigated and instructions will be worked out for applying the concept of decomposition in a concrete case. Their use and the benefit of the decomposition concept will be demonstrated at the special case G = SO(3).