  A weighted U-statistic based on a random sample X_1,...,X_n has the form
U_n=\sum_{1\le i,j\le n}w_{i-j}K(X_i,X_j), where K is a fixed symmetric
measurable function and the w_i 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 U_n when the sample
observations come from a nonlinear time series and linear processes.
