Distribution of angles between unit vectors and the multiple comparison problem for unit vectors

We suppose that L x i = Fq(μi, KiK) for i=1,...,p, where the independent Fisher distributions on the unit sphere in Rq have modal vectors μi and known concentrations KiK, where K will be large—in fact, to obtain asymptotic distributions, we will let K tend to infinity. The term x'iXj=cos 0ij estimates cos 0ij= μiμy. The asymptotic joint distribution of the terms will be studied when all the vectors μi μ are distinct, and for the special case when all the vectors μi=μ, so that all the terms are zero. These very different results have a variety of applications as well as being interesting in themselves. One of these applications is the 'multiple comparisons problem' for unit vectors. However, it was found necessary to give here a different way of solving this problem.