Power-law estimates for the central limit theorem for convex sets

By Bo'az Klartag. 2006

149 citations recorded by the source corpus.

Abstract We investigate the rate of convergence in the central limit theorem for convex sets established in [B. Klartag, A central limit theorem for convex sets, Invent. Math., in press. [8] ]. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9b21c908041e5db4026ec1839ac24f62bf269f43

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Approximately gaussian marginals and the hyperplane conjecture (2010, background): “41 where C > 0 is a universal constant (see [19]), and it is suggested in the works of Anttila, Ball and Perissinaki [1] and of Bobkov and Koldobsky [4] that perhaps σn ≤ C (3) for a universal constant C > 0.”
On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution (2009, background): “7 below that because of the concentration phenomenon, in particular, by important recent results of [17, 18] and also [12], such random vectors are good, as also are the vectors whose probability law is log-concave.”
A Polynomial Number of Random Points Does Not Determine the Volume of a Convex Body (2009, background): “pret these results as follows: provided a small number of points from a logarithmically-concave measure, one cannot distinguish it from a spherically symmetric measure. For deﬁnitions and results see [K]. One of the main stages of our proof is to show that one cannot distinguish between the uniform ”
On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries (2011, methodology): “|) obtained by Klartag [24], which will ensure that the condition (A3) is satisfied, allowing us to use the results of Section 2.”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, result): “Klartag’s improved estimate in [50] reads: V arμ(|x− Eμx|) ≤ Cεnσ1(μ) ∀ε > 0 .”
Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling (2009, methodology): “We will use here the following result immediately derived from [19], Theorem 4.”
Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures (2010, methodology): “Our main technical result in this work is the following improvement over the logLipschitz bounds of Klartag from [24]: Theorem 1.”
Bourgain’s slicing problem and KLS isoperimetry up to polylog (2022, methodology): “Earlier bounds for σn utilized the Concentration of Measure Phenomenon, see [31], Fleury [25] and Guédon and Milman [26], following Paouris’ large deviation principle [39].”
Small ball probability estimates for log-concave measures (2012, background): “Klartag in [12] for these measures (see also [7] for an alternative proof and [13], [6] for related developments).”
On Stein's method for multivariate normal approximation (2009, background): “It was recently shown by Bo’az Klartag in [13] (and quantitatively improved by Klartag in [14]) that in fact most such projections of a random vector drawn from a convex body are close to Gaussian, without any symmetry assumptions.”