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.
"Inference of weighted $V$-statistics for nonstationary time series and its applications." Ann. Statist. 42 (1) 87 - 114, February 2014. https://doi.org/10.1214/13-AOS1184