Probability Theory and Related Fields

, Volume 145, Issue 1, pp 1–33

A Berry-Esseen type inequality for convex bodies with an unconditional basis

Article

DOI: 10.1007/s00440-008-0158-6

Cite this article as:
Klartag, B. Probab. Theory Relat. Fields (2009) 145: 1. doi:10.1007/s00440-008-0158-6

Abstract

Suppose X = (X1, . . . , Xn) is a random vector, distributed uniformly in a convex body \({K \subset \mathbb R^n}\) . We assume the normalization \({\mathbb E X_i^2 = 1}\) for i = 1, . . . , n. The body K is further required to be invariant under coordinate reflections, that is, we assume that (±X1, . . . , ±Xn) has the same distribution as (X1, . . . , Xn) for any choice of signs. Then, we show that
$$ \mathbb E \left( \, |X| - \sqrt{n} \, \right)^2 \leq C^2,$$
where C  ≤  4 is a positive universal constant, and | · | is the standard Euclidean norm in \({\mathbb R^n}\) . The estimate is tight, up to the value of the constant. It leads to a Berry-Esseen type bound in the central limit theorem for unconditional convex bodies.

Mathematics Subject Classification (2000)

60F05 52A20 52A38 60D05 

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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