This is an announcement for the paper "L_p moments of random vectors via majorizing measures" by Olivier Guedon and Mark Rudelson.

Abstract: For a random vector X in R^n, 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.

Archive classification: Functional Analysis

Mathematics Subject Classification: 46B09, 52A21

Remarks: 32 pages, to appear in Advances in Mathematics

The source file(s), ADVgr06-03-15.tex: 71461 bytes, is(are) stored in gzipped form as 0507023.gz with size 21kb. The corresponding postcript file has gzipped size 108kb.

Submitted from: rudelson@math.missouri.edu

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