Projections of probability distributions: A measure-theoretic Dvoretzky theorem

Details Projections of probability distributions: A measure-theoretic Dvoretzky theorem Projections of probability distributions: A measure-theoretic Dvoretzky theorem

2011-02-16

arxiv.org/abs/1102.3438v3

Mathematics

Probability

Minor errors corrected resulting in cosmetic changes to statements of some theorems. Assumptions on the underlying distributio

Scientific paper

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.

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