The Uniform Distribution on a Sphere in ${\bf R}^S$. Properties of Projections. I

Abstract

The distribution law of the first k coordinates of a point uniformly distributed over a high dimensional sphere and the distribution law of k independent standard normal variables, as $n\to\infty$ with k fixed, are considered. The main result of this paper is a lower bound on the variational distance. The well‐known upper bound due to Diaconis and Freedman has been made more precise.