High dimensional non-Gaussian time series data are increasingly encountered in a wide range of applications. Conventional methods are inadequate for estimating mean vectors and second-order characteristics when it comes to ultra high dimensional and heavy tailed data. We adopt the framework of functional dependence measures and establish a Bernstein-type inequality under dependence. Then we study Huber estimation of means for high dimensional time series with the existence of (1 + )-th moments for some 0 < ≤ 1 and establish a phase transition for Huber estimators: it admits nearly sub-Gaussian concentration around the unknown mean for = 1 and a slower convergence rate if 0 < < 1. We also investigate Huber type estimators of covariance and precision matrices for the process with the existence of (2 + 2 )-th moments for some 0 < ≤ 1 and present the convergence rates for robust modification of regularized estimators. Similarly, a phase transition occurs between = 1 and 0 < < 1. As a significant improvement, the dimension can be allowed to increase exponentially with the sample size to ensure consistency under very mild moment conditions. Numerical results indicate a good performance of Huber estimates. Statistica Sinica: Newly accepted Paper (accepted author-version subject to English editing)

中文翻译：

---

高维时间序列均值和协方差矩阵的稳健估计

在广泛的应用中越来越多地遇到高维非高斯时间序列数据。当涉及超高维和重尾数据时，传统方法不足以估计均值向量和二阶特征。我们采用函数依赖度量的框架，并在依赖下建立伯恩斯坦型不等式。然后我们研究了高维时间序列均值的 Huber 估计，其中存在 (1 + )-th 矩对于某些 0 < ≤ 1 并建立 Huber 估计量的相变：它允许在未知均值附近接近亚高斯集中= 1 和较慢的收敛速度如果 0 < < 1。我们还研究了协方差和精度矩阵的 Huber 类型估计器，其中存在 (2 + 2 )-th 矩对于某些 0 < ≤ 1 并呈现正则化估计量的鲁棒修改的收敛速度。类似地，相变发生在 = 1 和 0 < < 1 之间。作为一项重大改进，可以允许维度随样本大小呈指数增加，以确保在非常温和的力矩条件下的一致性。数值结果表明 Huber 估计的性能良好。Statistica Sinica：新接受的论文（接受的作者版本需英文编辑）