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}
}
  • B. Klartag
  • Published 7 May 2007
  • Mathematics
  • Probability Theory and Related Fields
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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