A weighted U-statistic based on a random sample X_1,...,X_n has the form U_n=sum_{1le i,jle 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.
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