Abstract: High-dimensional nonGaussian time series data are becoming increasingly common. However, the conventional methods used to estimate mean vectors and second-order characteristics are inadequate for ultrahigh-dimensional and heavy-tailed data. Therefore, we use a framework of functional dependence measures to establish a Bernstein-type inequality under dependence. Then, we investigate a Huber estimator for the mean for a high-dimensional time series with (1 + ϵ)th moments, for some 0 < ϵ ≤ 1, and establish a phase transition for Huber estimators. The transition admits nearly subGaussian concentration around the unknown mean for ϵ = 1, and a slower convergence rate if 0 < ϵ < 1. We also investigate Huber-type estimators for the covariance and precision matrices of the process with (2 + 2ϵ)th moments, for some 0 < ϵ ≤ 1, and present the convergence rates for robust modifications of the regularized estimators. Similarly, a phase transition occurs between ϵ = 1 and 0 < ϵ < 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 that the Huber estimates perform well.

Key words and phrases: Bernstein-type inequality, covariance and precision matrix, heavy tailed data, high dimensional time series, Huber estimation, phase transition, regularized estimation.