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.

中文翻译：

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高维局部平稳过程的协方差和频谱密度估计的收敛性

协方差和频谱密度函数在时间序列理论中起着基本作用。对于低维平稳过程的估计，有一个完善的渐近理论。但是，对于高维非平稳过程，关于其渐近行为的许多重要问题仍未得到解答。本文提出了一种系统渐近理论，用于估计一般类别的高维局部平稳过程的时变二阶统计量。使用功能依赖性度量的框架，我们得出估计的收敛速度，该收敛速度取决于样本量$ T $，维数$ p $，矩条件和基础过程的依赖性。