A Berry-Esseen type inequality for convex bodies with an unconditional basis

We provide a sharp rate of convergence in the central limit theorem for random vectors with an unconditional, log-concave density. The argument relies on analysis of the Neumann laplacian on convex domains and on the theory of optimal transportation of measures.

💡 Research Summary

The paper addresses the quantitative central limit theorem (CLT) for high‑dimensional random vectors whose density is both log‑concave and unconditional with respect to a fixed orthogonal basis. An unconditional density is invariant under sign changes of any coordinate, which endows the distribution with a strong product‑type symmetry. The authors prove a Berry‑Esseen‑type inequality that gives a sharp rate of convergence of the normalized sum of the coordinates to the standard Gaussian law.

Main Result.
Let (X=(X_{1},\dots ,X_{n})) be a random vector in (\mathbb R^{n}) with an unconditional log‑concave density (f). Define the normalized sum
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