Projections of Probability Distributions: A Measure-Theoretic Dvoretzky Theorem

Abstract

Many authors have studied the phenomenon of typically Gaussian marginals of high-dimensional random vectors; e.g., for a probability measure on ${\mathbb{R}}^{d}$, under mild conditions, most one-dimensional marginals are approximately Gaussian if $d$ is large. In earlier work, the author used entropy techniques and Stein’s method to show that this phenomenon persists in the bounded-Lipschitz distance for $k$-dimensional marginals of d-dimensional distributions, if $k = o(\sqrt{\log (d)})$. In this paper, a somewhat different approach is used to show that the phenomenon persists if $k < \frac{2\log (d)} {\log (\log (d))}$, and that this estimate is best possible.