On the Characteristic Function of the Matrix von Mises—Fisher Distribution with Application to SO(N)—Deconvolution

Abstract

Characteristic functions provide a useful method for analyzing probability distributions. In most instances, the domain is taken to be Euclidean space and although the integral transforms may not have a simple expression, some qualitative features about the underlying probability distribution can often be extracted. Although there is some research efforts in spaces more complicated then the Euclidean case, very little is available on characteristic functions for probability distributions on these general domains. In this paper we will calculate the characteristic function of a probability distribution on a more general domain, in particular, the SO(N)—version of the matrix von Mises-Fisher distribution. It is found that when Fourier transforms of the latter are taken with respect to the irreducible representations of SO(N), a concrete expression for the characteristic function is obtained. In addition, some qualitative properties are found which in turn are used to obtain L 2—rates of convergence for the SO(N)—deconvolution problem.