Abstract
We prove optimal convergence rates in the central limit theorem for sums ${\bf R}^k.$ Assuming a fourth moment, we obtain a Berry-Esseen type bound of $O(N^{-1})$ for the probability of hitting a ball provided that $k\leq 5$. The proof still requires a technical assumption related to the independence of coordinate sums.
Citation
V. Bentkus. F. Götze. "Optimal rates of convergence in the CLT for quadratic forms." Ann. Probab. 24 (1) 466 - 490, January 1996. https://doi.org/10.1214/aop/1042644727
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