Under correlation-type conditions, we derive an upper bound of order (log n)/n for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.
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Acknowledgments. The authors would like to thank the two referees for careful reading of the manuscript and valuable comments. This work was supported by NSF Grant DMS-1855575, the Simons Foundation and CRC 1283 at Bielefeld University.
- Normal approximation
- Sudakov's typical distributions