  We investigate the behavior of Fourier transforms for a wide class of
nonstationary nonlinear processes. Asymptotic central and noncentral limit
theorems are established for a class of nondegenerate and degenerate weighted
$V$-statistics through the angle of Fourier analysis. The established theory
for $V$-statistics provides a unified treatment for many important time and
spectral domain problems in the analysis of nonstationary time series, ranging
from nonparametric estimation to the inference of periodograms and spectral
densities.
