Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces

Abstract

If X = (X₁, · ··, X_n) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.