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Projections of Probability Distributions: A Measure-Theoretic Dvoretzky Theorem

  • Elizabeth MeckesEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

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.

Keywords

Lipschitz Constant Total Variation Distance Standard Gaussian Measure Euclidean Subspace Gaussian Marginal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The author thanks Mark Meckes for many useful discussions, without which this paper may never have been completed. Thanks also to Michel Talagrand, who pointed out a simplification in the proof of the main theorem, and to Richard Dudley for clarifying the history of “Dudley’s entropy bound”. Research supported by an American Institute of Mathematics 5-year Fellowship and NSF grant DMS-0852898.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA

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