Convergence Rates for $U$-Statistics and Related Statistics

Abstract

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.