Maximal inequalities and convergence results for generalized U- statistics
The authors present some key probability inequalities and convergence results for multisample, or `generalized U-statistics'. They exploit the reverse martingale structure of generalized U-statistics to obtain a useful Kolmogorov-type inequality. In addition they obtain a bound for the rate of convergence in the strong law of large numbers for generalized U-statistics, extending the one-sample result of \textit{W. F. Grams} and \textit{R. J. Serfling} [Ann. of Statist. 1, 153-160 (1973; Zbl 0322.62053)]. Like \textit{P. K. Sen} [Ann. Probab. 5, 287-290 (1977; Zbl 0362.60019 )] they exhibit another class of generalized U-statistics for which the strong law holds under merely a first moment condition. Finally using \textit{W. Hoeffding}'s [The strong law of large numbers for U-statistics. Univ. North Carolina Inst. Stat. Mimeo Ser. No.302 (1961)] forward martingale representation they establish an invariance principle for generalized U-statistics.
scientific article; zbMATH DE number 4135207
scientific article