Weak invariance principles for weightedU-statistics

We prove that the distribution of a properly normalized weightedU-statisticU_n in i.i.d. random variables is close to the distribution of a certain functionV_n in i.i.d. standardized Gaussian random variables in the sense that their Lévy-Prokhorov distance tends to zero asn. This property is then used to determine the limit laws ofU_n under special assumptions on the kernel function. This generalizes a method due to Rotar' who proved similar results for random multilinear forms.