A weighted U-statistic based on a random sample X1,…,Xn has the form Un=∑1≤i,j≤nwi−jK(Xi,Xj), where K is a fixed symmetric measurable function and the wi are symmetric weights. A large class of statistics can be expressed as weighted U-statistics or variations thereof. This paper establishes the asymptotic normality of Un when the sample observations come from a nonlinear time series and linear processes.
"On weighted U-statistics for stationary processes." Ann. Probab. 32 (2) 1600 - 1631, April 2004. https://doi.org/10.1214/009117904000000333