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… 

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