Spectral Inference for High Dimensional Time Series

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.