Projections of Probability Distributions: A Measure-Theoretic Dvoretzky Theorem

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


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.


Lipschitz Constant Total Variation Distance Standard Gaussian Measure Euclidean Subspace Gaussian Marginal 
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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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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA

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