Covariances and spectral density functions play a fundamental role in the theory of time series. There is a well-developed asymptotic theory for their estimates for low-dimensional stationary processes. For high-dimensional nonstationary processes, however, many important problems on their asymptotic behaviors are still unanswered. This paper presents a systematic asymptotic theory for the estimates of time-varying second-order statistics for a general class of high-dimensional locally stationary processes. Using the framework of functional dependence measure, we derive convergence rates of the estimates which depend on the sample size $T$, the dimension $p$, the moment condition and the dependence of the underlying processes.
"Convergence of covariance and spectral density estimates for high-dimensional locally stationary processes." Ann. Statist. 49 (1) 233 - 254, February 2021. https://doi.org/10.1214/20-AOS1954