On the Berry-Esseen Theorem for $U$-Statistics
On the Berry-Esseen Theorem for $U$-Statistics
Assuming the existence of fourth moment only, we prove the Berry-Esseen theorem for $U$-statistics. Assuming the third absolute moment, we obtain the order bound $O(n^{-\frac{1}{2}} \log^{\frac{1}{3}}n)$. This improves earlier results of Bickel, and Grams and Serfling.