Approximation of Projections of Random Vectors

Let X be a d-dimensional random vector and X θ its projection onto the span of a set of orthonormal vectors {θ 1,…,θ k }. Conditions on the distribution of X are given such that if θ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from X θ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of d, k, and the distribution of X, allowing consideration not just of fixed k but of k growing with d. The results are applied in the setting of projection pursuit, showing that most k-dimensional projections of n data points in ℝ d are close to Gaussian, when n and d are large and k=clog (d) for a small constant c.