, Volume 168, Issue 1, pp 91-131
Date: 10 Jan 2007

A central limit theorem for convex sets

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Abstract

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.