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.

Minor errors corrected resulting in cosmetic changes to statements of some theorems. Assumptions on the underlying distribution have been clarified; a small amount of additional discussion added