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August 2021 A new look at random projections of the cube and general product measures
Zakhar Kabluchko, Joscha Prochno, Christoph Thäle
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Bernoulli 27(3): 2117-2138 (August 2021). DOI: 10.3150/20-BEJ1303

Abstract

A consequence of the celebrated Dvoretzky–Milman theorem is a strong law of large numbers for d-dimensional random projections of the n-dimensional cube. It shows that, with respect to the Hausdorff distance, a uniform random projection of the cube [1/n,+1/n]n onto Rd converges almost surely to a centered d-dimensional Euclidean ball of radius 2/π, as n. We start by providing an alternative proof of this strong law via the Artstein–Vitale law of large numbers for random compact sets. Then, for every point inside the ball of radius 2/π, we determine the asymptotic number of vertices and the volume of the part of the cube projected ‘close’ to this point. More generally, we study large deviations for random projections of arbitrary product measures. Let νn be the n-fold product measure of a Borel probability measure ν on R, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in Rn. It is shown that the sequence of random measures νn(n1/2I)1, nN, satisfies a large deviation principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of ν. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube [1,1]n and the discrete cube {1,1}n as special cases.

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Zakhar Kabluchko. Joscha Prochno. Christoph Thäle. "A new look at random projections of the cube and general product measures." Bernoulli 27 (3) 2117 - 2138, August 2021. https://doi.org/10.3150/20-BEJ1303

Information

Received: 1 September 2020; Revised: 1 November 2020; Published: August 2021
First available in Project Euclid: 10 May 2021

Digital Object Identifier: 10.3150/20-BEJ1303

Rights: Copyright © 2021 ISI/BS

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Vol.27 • No. 3 • August 2021
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