A Concentration Theorem for Projections

Suppose the random vector X ∈ R has mean zero and finite second moments. We show that there is a precise sense in which almost all linear projections of X into R (for d < D) look like a scale-mixture of spherical Gaussians—specifically, a mixture of distributions N(0, σ2Id) where the σ values follow the same distribution as ‖X‖/ √ D. The extent of this effect depends upon the ratio of d to D, and upon a particular coefficient of eccentricity of X’s distribution.