, Volume 145, Issue 1-2, pp 1-33
Date: 07 Jun 2008

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

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Abstract

Suppose X = (X 1, . . . , X n ) 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 (±X 1, . . . , ±X n ) has the same distribution as (X 1, . . . , X n ) 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.

The author is a Clay Research Fellow, and is also supported by NSF grant #DMS − 0456590.