A new look at random projections of the cube and general product measures

Abstract

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 ${[-1/\sqrt{\mathit{n}},+1/\sqrt{\mathit{n}}]}^{\mathit{n}}$ onto ${\mathbb{R}}^{\mathit{d}}$ converges almost surely to a centered d-dimensional Euclidean ball of radius $\sqrt{2/\mathit{\pi }}$, as $\mathit{n}\to \infty$. 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 $\sqrt{2/\mathit{\pi }}$, 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 ${\mathit{\nu }}^{\otimes \mathit{n}}$ be the n-fold product measure of a Borel probability measure ν on $\mathbb{R}$, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in ${\mathbb{R}}^{\mathit{n}}$. It is shown that the sequence of random measures ${\mathit{\nu }}^{\otimes \mathit{n}}\circ {({\mathit{n}}^{-1/2}{\mathit{I}}^{\ast })}^{-1}$, $\mathit{n}\in \mathbb{N}$, satisfies a large deviation principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of ν. 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 ${[-1,1]}^{\mathit{n}}$ and the discrete cube ${\{-1,1\}}^{\mathit{n}}$ as special cases.