A Bernstein-type Inequality for High Dimensional Linear Processes with Applications to Robust Estimation of Time Series Regressions

TL;DR

This paper introduces a new Bernstein-type inequality tailored for high-dimensional linear processes, enabling robust estimation in complex time series models with fat tails and serial dependence.

Contribution

It develops a novel Bernstein-type inequality for high-dimensional linear processes and applies it to improve robust estimation methods in high-dimensional time series analysis.

Findings

Allows exponential dimension growth with sample size

Ensures consistency in high-dimensional robust estimation

Applicable to fat-tailed, correlated data in time series

Abstract

Time series regression models are commonly used in time series analysis. However, in modern real-world applications, serially correlated data with an ultra-high dimension and fat tails are prevalent. This presents a challenge in developing new statistical tools for time series analysis. In this paper, we propose a novel Bernstein-type inequality for high-dimensional linear processes and apply it to investigate two high-dimensional robust estimation problems: (1) time series regression with fat-tailed and correlated covariates and errors, and (2) fat-tailed vector autoregression. Our proposed approach allows for exponential increases in dimension with sample size under mild moment and dependence conditions, while ensuring consistency in the estimation process.