TY - JOUR

T1 - Pointwise estimates for marginals of convex bodies

AU - Eldan, R.

AU - Klartag, B.

N1 - Funding Information:
* Corresponding author. E-mail addresses: roneneldan@gmail.com (R. Eldan), bklartag@princeton.edu (B. Klartag). 1 Partly supported by the Israel Science Foundation. 2 Supported by the Clay Mathematics Institute and by NSF grant #DMS-0456590.

PY - 2008/4/15

Y1 - 2008/4/15

N2 - We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let μ be an isotropic, log-concave probability measure on Rn. For a typical subspace E ⊂ Rn of dimension nc, consider the probability density of the projection of μ onto E. We show that the ratio between this probability density and the standard Gaussian density in E is very close to 1 in large parts of E. Here c > 0 is a universal constant. This complements a recent result by the second named author, where the total variation metric between the densities was considered.

AB - We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let μ be an isotropic, log-concave probability measure on Rn. For a typical subspace E ⊂ Rn of dimension nc, consider the probability density of the projection of μ onto E. We show that the ratio between this probability density and the standard Gaussian density in E is very close to 1 in large parts of E. Here c > 0 is a universal constant. This complements a recent result by the second named author, where the total variation metric between the densities was considered.

KW - Central limit theorem

KW - Convex bodies

KW - Marginal distribution

UR - http://www.scopus.com/inward/record.url?scp=39849097676&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2007.08.014

DO - 10.1016/j.jfa.2007.08.014

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AN - SCOPUS:39849097676

VL - 254

SP - 2275

EP - 2293

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

ER -