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

@article{Klartag2007ABT, title={A Berry-Esseen type inequality for convex bodies with an unconditional basis}, author={Bo'az Klartag}, journal={Probability Theory and Related Fields}, year={2007}, volume={145}, pages={1-33} }

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…

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