Optimal rates of convergence in the CLT for quadratic forms

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.