Projections of probability distributions: A measure-theoretic Dvoretzky theorem

TL;DR

This paper extends the understanding of high-dimensional probability distributions by establishing a measure-theoretic Dvoretzky theorem, showing that most low-dimensional projections of high-dimensional measures are approximately Gaussian, with improved bounds.

Contribution

The paper introduces a new approach to prove that most low-dimensional marginals of high-dimensional distributions are approximately Gaussian, extending previous bounds and establishing their optimality.

Findings

Most low-dimensional marginals are approximately Gaussian.

The bound for the dimension of marginals is improved to $k<\frac{2\log(d)}{\log(\log(d))}$.

The new bound is proven to be optimal.

Abstract

Many authors have studied the phenomenon of typically Gaussian marginals of high-dimensional random vectors; e.g., for a probability measure on $\R^d$, 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.