A central limit theorem for convex sets

By Bo'az Klartag. 2006

262 citations recorded by the source corpus.

We show that there exists a sequence $\varepsilon_n\searrow0$ for which the following holds: Let K⊂ℝn be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝn, t0∈ℝ and σ>0 such that $$\sup_{A\subset\mathbb{R}}\left|\textit{Prob}\,\{\langle X,\theta\rangle\in A\}-\frac{1}{\sqrt{2\pi\sigma}}\int_Ae^{-\frac{(t - t_0)^2}{2\sigma^2}} dt\right|\leq\varepsilon_n,\qquad{(\ast)}$$ where the supremum runs over all measurable sets A⊂ℝ, and where 〈·,·〉 denotes the usual scalar product in ℝn. Furthermore, under the additional assumptions that the expectation of X is zero and that the covariance matrix of X is the identity matrix, we may assert that most unit vectors θ satisfy (*), with t0=0 and σ=1. Corresponding principles also hold for multi-dimensional marginal distributions of convex sets.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c50b8614e060761e6f0918cbf7b14c82b425fb02

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Citations in the corpus

Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures (2010, background): “Klartag in [23], involving delicate logarithmic improvements in n over the trivial bounds.”
Robustly Learning a Gaussian: Getting Optimal Error, Efficiently (2017, background): “Furthermore, a random projection will look Gaussian with high probability [Kla07].”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, result): “I am also grateful to Franck Barthe for his kind hospitality, remarks and advice, Bo’az Klartag for several references, discussions and insightful remarks, Professors Sergey Bobkov and Michel Ledoux for their comments and advice, and Professor David Jerison for several interesting conversations and ”
Bourgain’s slicing problem and KLS isoperimetry up to polylog (2022, methodology): “The thin-shell constant is crucial for establishing the Central Limit Theorem for Convex Sets [30, 32], as put forth in [1] following Sudakov [40] and Diaconis and Freedman [19].”
Approximately gaussian marginals and the hyperplane conjecture (2010, background): “The importance of the parameter σn stems from the central limit theorem for convex bodies [18].”
Relative entropy of cone measures and Lp centroid bodies (2009, methodology): “We make use of the following lemma due to Klartag [24] (Lemmas 4.3 and 4.5).”
High-dimensional Bayesian inference via the unadjusted Langevin algorithm (2016, methodology): “Therefore, using [22, Lemma 4.9] (or the Pinsker inequality), we get the following upper bound: , where C is a universal constant.”
A Berry-Esseen type inequality for convex bodies with an unconditional basis (2007, methodology): “Previous techniques for obtaining thin spherical shell est imates under convexity assumptions relied almost entirely on concentration of measure ideas, either on the sphere (see [15, 22]), or on the orthogonal group (see [23 ]).”
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.”
On the infimum convolution inequality (2008, background): “3 we show how IC implies a comparison of weak and strong moments and the results of [12] and [18].”
Power-law estimates for the central limit theorem for convex sets (2006, methodology): “The rest of the argument is similar to the proof of Theorem 1.4 in [9]; see the derivation involving formulas (49), (50) in [ 9, Section 4 ]. We have proven that (48) holds for all E ∈ E. Recall that σn,k(E) > 1 − C exp(−cnλ).”
A stability result for mean width of Lp-centroid bodies (2007, background): “This result has been very recently obtained in full generality by Klartag [12], where he proved that (1) holds true with 2e−ε 2 log n for every isotropic convex body with center of mass at the origin.”