On the invariance principle for U-statistics

Let Tn be a U-statistic and Sn its projection (in the sense of Hájek). Limit theory for U-statistics is usually considered in two disjoint cases, termed degenerate and nondegenerate. The traditional method is to treat the cases separately, using different techniques in each to obtain a solution. Here we present a unified treatment based on a joint invariance principle for the vector (Tn, Tn - Sn), from which the invariance principles in both the degenerate and nondegenerate cases follow as easy corollaries.