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A Berry-Esseen type inequality for convex bodies with an unconditional basis
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  • Published: 07 June 2008

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

  • Bo’az Klartag1 

Probability Theory and Related Fields volume 145, pages 1–33 (2009)Cite this article

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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.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Bo’az Klartag

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  1. Bo’az Klartag
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Correspondence to Bo’az Klartag.

Additional information

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

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Klartag, B. A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Relat. Fields 145, 1–33 (2009). https://doi.org/10.1007/s00440-008-0158-6

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  • Received: 07 May 2007

  • Revised: 12 April 2008

  • Published: 07 June 2008

  • Issue Date: September 2009

  • DOI: https://doi.org/10.1007/s00440-008-0158-6

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Mathematics Subject Classification (2000)

  • 60F05
  • 52A20
  • 52A38
  • 60D05
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