Assuming only the existence of the third absolute moment we prove that ₓ |P(σₙ⁻¹ Uₙ x) - Φ (x)| Cν₃σg⁻³n-1/2 where Uₙ is a U-statistic. This concludes a series of investigations on the Berry-Esseen theorem for U-statistics by Grams and Serfling, Bickel, and Chan and Wierman.
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Callaert et al. (1978) studied this question.