• DOI:10.1007/s00222-006-0028-8
• Corpus ID: 119169773

A central limit theorem for convex sets

@article{Klartag2007ACL,
  title={A central limit theorem for convex sets},
  author={Bo'az Klartag},
  journal={Inventiones mathematicae},
  year={2007},
  volume={168},
  pages={91-131}
}

• B. Klartag
• Published 29 April 2006
• Mathematics
• Inventiones mathematicae

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.