This is an announcement for the paper "Combinatorics of random processes and sections of convex bodies" by Mark Rudelson and Roman Vershynin.

Abstract: We find a sharp combinatorial bound for the metric entropy of sets in R^n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in R^n. In the operator theory, this essentially proves for all normed spaces the restricted invertibility principle of Bourgain and Tzafriri.

Archive classification: Functional Analysis; Probability Theory

Mathematics Subject Classification: 46B09, 60G15, 68Q15

Remarks: 49 pages

The source file(s), rv-processes.tex: 122610 bytes, is(are) stored in gzipped form as 0404192.gz with size 38kb. The corresponding postcript file has gzipped size 150kb.

Submitted from: vershynin@math.ucdavis.edu

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http://arXiv.org/abs/math.FA/0404192

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