  High-dimensional linear regression has been intensively studied in the
community of statistics in the last two decades. For the convenience of
theoretical analyses, classical methods usually assume independent observations
and sub-Gaussian-tailed errors. However, neither of them hold in many real
high-dimensional time-series data. Recently [Sun, Zhou, Fan, 2019, J. Amer.
Stat. Assoc., in press] proposed Adaptive Huber Regression (AHR) to address the
issue of heavy-tailed errors. They discover that the robustification parameter
of the Huber loss should adapt to the sample size, the dimensionality, and the
moments of the heavy-tailed errors. We progress in a vertical direction and
justify AHR on dependent observations. Specifically, we consider an important
dependence structure -- Markov dependence. Our results show that the Markov
dependence impacts on the adaption of the robustification parameter and the
estimation of regression coefficients in the way that the sample size should be
discounted by a factor depending on the spectral gap of the underlying Markov
chain.
