Optimal rates of convergence in the CLT for quadratic forms

Bentkus V, Götze F. Optimal rates of convergence in the CLT for quadratic forms. ANNALS OF PROBABILITY. 1996;24(1):466-490.We prove optimal convergence rates in the central limit theorem for sums in 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 greater than or equal to 5. The proof still requires a technical assumption related to the independence of coordinates of sums