L_p moments of random vectors via majorizing measures

2005/07/01 by Olivier Guedon, Guedon, Olivier, Mark Rudelson +1
Mathematics · #46B09 #52A21 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B09 #msc:52A21

paper · pdf · doi:10.48550/arxiv.math/0507023

32 pages, to appear in Advances in Mathematics

arxiv created 2006/04/04 · arxiv updated 2009/12/01

Abstract

For a random vector X in Rⁿ, we obtain bounds on the size of a sample, for which the empirical p-th moments of linear functionals are close to the exact ones uniformly on an n-dimensional convex body K. We prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis. In particular, we find a short Lewis type decomposition for any finite dimensional subspace of L_p. We also prove that for an isotropic log-concave random vector, we only need about n^p/2 log n sample points so that the empirical p-th moments of the linear functionals are almost isometrically the same as the exact ones. We obtain a concentration estimate for the empirical moments. The main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain Gaussian process.

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