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Inst. , Harris, J. (1991): Representation Theory. A First Course. Graduate Texts in Mathematics, 129, Springer-Verlag, New York [P] Pisier, G. (1989): The Volume of Convex Bodies and Banach Space Geometry. G. Bobkov School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. edu Summary. For non-correlated random variables, we study a concentration property of the distributions of the weighted sums with Bernoullian coefficients. The obtained result is used to derive an “almost surely version” of the central limit theorem.
The proof might require some information on the distribution of the Euclidean norm of a point x over K. Indeed, if we observe x = (x1 , . . , xn ) as a random vector uniformly distributed in K, and if (ε1 , . . , εn ) is an arbitrary collection of signs, then (ε1 x1 , . . , εn xn ) has the same uniform distribution (by the assumption that the canonical basis is unconditional). In particular, f (x, ε) = ε1 x1 + . . + εn xn √ n has the same distribution as f (x). But with respect to the symmetric Bernoulli measure Pε on the discrete cube {−1, 1}n , there is a subgaussian inequality 2 2 Pε |f (x, ε)| ≥ t ≤ 2 e−nt /(2|x| ) , t ≥ 0.
We would like to thank V. D. Milman for stimulating discussions. References [A] [A-B-P] [Ba] [B-P] [Bob] Alesker, S. (1995): ψ2 -estimate for the Euclidean norm on a convex body in isotropic position. Geom. Aspects Funct. Anal. (Israel 19921994), Oper. Theory Adv. , Perissinaki, I. (1998): The central limit problem for convex bodies. Preprint Ball, K. (1988): Logarithmically concave functions and sections of convex sets. , Perissinaki, I. (1998): Subindependence of coordinate slabs in n p balls.