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

Abstract

Suppose X = (X 1, . . . , X n ) is a random vector, distributed uniformly in a convex body K ⊂ ℝn. We assume the normalization EXi2 = 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 E (|X| - n,)2 ≤ Cn where C ≤ 4 is a positive universal constant, and | • | is the standard Euclidean norm in ℝn. 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. © 2008 Springer-Verlag.