On weighted U-statistics for stationary processes

Abstract

A weighted U-statistic based on a random sample X₁,…,X_n has the form U_n=∑_1≤i,j≤nw_i−jK(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.