Convergence of covariance and spectral density estimates for high-dimensional locally stationary processes
The Annals of Statistics · 2021 · Volume 49 · Issue 1
Abstract
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.
Keywords
62M10, 62M15, convergence rate, Hanson–Wright-type inequalities, high-dimensional time series, Locally stationary processes, second-order statistics
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