TAILIEUCHUNG - Đề tài " Combinatorics of random processes and sections of convex bodies "

We find a sharp combinatorial bound for the metric entropy of sets in Rn 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 Rn . . | Annals of Mathematics Combinatorics of random processes and sections of convex bodies By M. Rudelson and R. Vershynin Annals of Mathematics 164 2006 603 648 Combinatorics of random processes and sections of convex bodies By M. Rudelson and R. VERSHyNiN Abstract We find a sharp combinatorial bound for the metric entropy of sets in Rra 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 Rra. In the operator theory this essentially proves for all normed spaces the restricted invertibility principle of Bourgain and Tzafriri. 1. Introduction This paper develops a sharp combinatorial method for estimating metric entropy of sets in Rra and equivalently of function classes on a probability space. A need in such estimates occurs naturally in a number of problems of analysis functional harmonic and approximation theory probability combinatorics convex and discrete geometry statistical learning theory etc. Our entropy method which evolved from the work of Mendelson and the second author MV 03 is motivated by several problems in the empirical processes asymptotic convex geometry and operator theory. Throughout the paper F is a class of real-valued functions on some domain Q. It is a central problem of the theory of empirical processes to determine whether the classical limit theorems hold uniformly over F. Let ụ. be a probability distribution on Q and X1 X2 . E Q be independent samples distributed according to a common law J. The problem is to determine whether the sequence of real-valued random variables f Xi obeys the central limit Research of . supported in part by NSF grant DMS-0245380. Research of

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