Bounds are provided for the rates of convergence in the central limit theorem and the strong law of large numbers for $U$-statistics. The results are obtained by establishing suitable bounds upon the moments of the difference between a $U$-statistic and its projection. Analogous conclusions for the associated von Mises statistical functions are indicated. Statistics considered for exemplification are the sample variance and the Wilcoxon two-sample statistic.
"Convergence Rates for $U$-Statistics and Related Statistics." Ann. Statist. 1 (1) 153 - 160, January, 1973. https://doi.org/10.1214/aos/1193342392