We consider estimation of high-dimensional long-run covariance matrices for time series with nonconstant means, a setting in which conventional estimators can be severely biased. To address this difficulty, we propose a difference-based initial estimator that is robust to a broad class of mean variations, and combine it with hard thresholding, soft thresholding, and tapering to obtain sparse long-run covariance estimators for high-dimensional data. We derive convergence rates for the resulting estimators under general temporal dependence and time-varying mean structures, showing explicitly how the rates depend on covariance sparsity, mean variation, dimension, and sample size. Numerical experiments show that the proposed methods perform favorably in high dimensions, especially when the mean evolves over time.

翻译：本文研究具有非恒定均值的时间序列的高维长程协方差矩阵估计问题，在此设定下传统估计量可能产生严重偏差。为解决此困难，我们提出一种基于差分的初始估计量，该估计量对广泛类别的均值变化具有鲁棒性，并将其与硬阈值化、软阈值化及锥化方法相结合，以获得适用于高维数据的稀疏长程协方差估计量。我们在一般时间依赖性与时变均值结构下推导了所得估计量的收敛速率，明确揭示了速率如何依赖于协方差稀疏性、均值变化、维度与样本量。数值实验表明，所提方法在高维情形下表现优异，尤其在均值随时间演化时更具优势。