For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance between the infinity norms of the estimation error of sample covariance matrix and the corresponding Gaussian approximation. The method utilizes martingale and m-dependent approximation and relies on constructing triadic blocks. We also establish a bootstrapping result with block sampling method, which preserves validity despite strong temporal dependence. Our results on covariance allow ultra-high-dimensional settings where the dimension of time series can grow sub-exponentially with sample size. Similar results can be built for precision matrix under low-dimensional settings. No assumption is required on the structure of covariance and precision matrices.

翻译：针对具有长程时间依赖性的时间序列，其协方差与精度矩阵的推断问题具有显著挑战性。我们提出一种Berry-Esseen型高斯近似结论，该结论给出了样本协方差矩阵估计误差无穷范数与相应高斯近似之间Kolmogorov距离的有限样本界。该方法利用鞅与m-相依逼近技术，并基于三元组区块的构造实现。同时，我们建立了基于区块采样法的自助法结果，该方法在强时间依赖性下仍保持有效性。关于协方差矩阵的结论可适用于超高维场景，其中时间序列维度可随样本量亚指数增长。在低维设定下，类似结论可推广至精度矩阵。本方法无需对协方差与精度矩阵的结构施加任何假设。