Spectral analysis plays a fundamental role in the study of time series. While there is a well-developed asymptotic theory for spectral density estimate in low-dimensional cases, a corresponding distributional theory for high-dimensional time series is still lacking. This paper aims to fill this gap by introducing a comprehensive inference theory for the spectral density estimate of high-dimensional time series with possibly nonlinear generating systems and non-Gaussian distributions. Our result is built across different dimensions and frequencies and can serve as a versatile tool for addressing various time series inference challenges. Additionally, we present two distinct resampling methods aimed at practical implementation of high-dimensional spectral inference, each accompanied by a theoretical justification of its validity.

High-order cumulants and high-order spectra play an important role in the theory of stationary processes. For nonlinear processes, it is quite challenging to develop an asymptotic theory for their estimators. This paper presents a systematic asymptotic theory for estimators of high-order moments for a general class of stationary processes, using the framework of functional dependence measures. In particular, we prove the asymptotic normality of the estimators and establish a uniform convergence rate. We also provide a sufficient condition for the summability of high-order cumulants. Based on the latter, we prove consistency and asymptotic normality of the third-order spectra or bispectrum estimators under mild and easily verifiable conditions.