Introduction

Abstract

The idea of studying probability measures on spheres in Euclidean space ℝ^p rather than on the Euclidean space itself is as old as the beginnings of probability theory and statistics. In 1734 Daniel Bernoulli looked at the orbital planes of the planes known at his time as random points on the surface of a sphere and asserted their uniform distribution. In the first quarter of this century Rayleigh and Karl Pearson started investigations on the resultant length of normal vectors, in connection with approximation problems for large samples, within the framework of random walks on spheres. Until then the distributions appearing in the work of the pioneers were all uniform.