Abstract: We present a general triangular array central limit theorem for U-statistics, where the kernel hk(x₁, . . . , x_k) and its dimension k may increase with the sample size. Motivating examples that require such a general result are presented, including a class of Hodges-Lehmann estimators, subsampling estimators, and combining p-values using data splitting. A result for the so-called M-statistic is also presented, which is defined as the median of some kernel computed over all subsets of the data of a given size. The conditions in the theorems are verified in the motivating examples as well.

Key words and phrases: Data splitting, Hodges-Lehmann estimator, hypothesis testing, P-values, subsampling, U-statistics.