Asymptotic distribution of coordinates on high dimensional spheres

Abstract

The coordinates xi of a point x = (x1, x2., xn) chosen at random according to a uniform distribution on the l2(n)-sphere of radius n1/2 have approximately a normal distribution when n is large. The coordinates xi of points uniformly distributed on the l1(n)-sphere of radius n have approximately a double exponential distribution. In these and all the lp(n), 1 ≤ p ≤ ∞, convergence of the distribution of coordinates as the dimension n increases is at the rate √n and is described precisely in terms of weak convergence of a normalized empirical process to a limiting Gaussian process, the sum of a Brownian bridge and a simple normal process. © 2007 Applied Probability Trust.