This paper concerns the minimax estimation of covariance and precision matrices for high-dimensional time series with long-memory property. We generalize the minimax results for the convergence rates of the estimation of covariance matrices in Shu and Nan (2019) in several directions with a mild assumption, which was mentioned as an open problem in Supplement to Cai and Zhou (2012) for i.i.d. data. We also obtain the minimax results for the convergence rates of the estimation of precision matrices under various norms, which is not considered by Shu and Nan (2019) and Cai and Zhou (2012).

中文翻译：

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长记忆高维时间序列协方差和精度矩阵的极大极小估计

本文涉及具有长记忆特性的高维时间序列的协方差和精度矩阵的极大极小估计。我们用温和的假设在几个方向上概括了 Shu and Nan (2019) 协方差矩阵估计收敛率的极大极小值结果，这在对 iid 数据的 Cai 和 Zhou (2012) 的补充中被提及为一个开放问题。我们还获得了各种范数下精度矩阵估计的收敛速度的极大极小值结果，Shu and Nan (2019) 和 Cai and Zhou (2012) 没有考虑到这一点。