The limiting distribution of weighted $U$-statistics of degree 2 is found for a wide class of weights, including uniform weights. Nonnormal limits can occur for both degenerate and nondegenerate kernels. A compact expression is given for the cumulants of the distribution. Incomplete and randomly weighted $U$-statistics are also analyzed.
"Asymptotic Distributions of Weighted $U$-Statistics of Degree 2." Ann. Probab. 21 (2) 1159 - 1169, April, 1993. https://doi.org/10.1214/aop/1176989286