Moment bounds for large autocovariance matrices under dependence

TL;DR

This paper derives expectation bounds for deviations of large autocovariance matrices under weak dependence, advancing understanding in high-dimensional time series analysis and applicable models.

Contribution

It establishes deviation bounds for autocovariance matrices under dependence, extending covariance estimation theory to dependent data scenarios.

Findings

Derived bounds depend on data's intrinsic dimension and logarithmic factors.

Results applicable to high-dimensional VAR and ARCH models.

Provides theoretical foundation for autocovariance estimation under dependence.

Abstract

The goal of this paper is to obtain expectation bounds for the deviation of large sample autocovariance matrices from their means under weak data dependence. While the accuracy of covariance matrix estimation corresponding to independent data has been well understood, much less is known in the case of dependent data. We make a step towards filling this gap, and establish deviation bounds that depend only on the parameters controlling the "intrinsic dimension" of the data up to some logarithmic terms. Our results have immediate impacts on high dimensional time series analysis, and we apply them to high dimensional linear VAR($d$) model, vector-valued ARCH model, and a model used in Banna et al. (2016).