A New Look At Random Projections Of The Cube And General Product Measures

A consequence of the celebrated Dvoretzky-Milman theorem is a strong law of large numbers for d-dimensional random projections of the n-dimensional cube. It shows that, with respect to the Hausdorff distance, a uniform random projection of the cube [-1/root n, +1/root n](n) onto R-d converges almost surely to a centered d-dimensional Euclidean ball of radius root 2/pi, as n -> infinity. We start by providing an alternative proof of this strong law via the Artstein-Vitale law of large numbers for random compact sets. Then, for every point inside the ball of radius root 2/pi, we determine the asymptotic number of vertices and the volume of the part of the cube projected `close' to this point. More generally, we study large deviations for random projections of arbitrary product measures. Let nu(circle times n) be the n-fold product measure of a Borel probability measure nu on R, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in R-n. It is shown that the sequence of random measures nu(circle times n) omicron(n(-1/2)I*)(-1), n is an element of N, satisfies a large deviation principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of nu. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube [-1, 1](n) and the discrete cube {-1, 1}(n) as special cases.