A new look at random projections of the cube and general product measures

نویسندگان

چکیده

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

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ژورنال

عنوان ژورنال: Bernoulli

سال: 2021

ISSN: ['1573-9759', '1350-7265']

DOI: https://doi.org/10.3150/20-bej1303