Simultaneous Inference for Covariance and Precision Matrices of Long-Range Dependent Time Series

TL;DR

This paper develops a Gaussian approximation and bootstrap method for inference on covariance and precision matrices in long-range dependent time series, applicable to ultra-high-dimensional settings.

Contribution

It introduces a Berry-Esseen type bound and a bootstrap procedure that work under strong temporal dependence without structural assumptions.

Findings

Finite-sample Gaussian approximation bounds for covariance matrix errors

Bootstrap method preserves validity under long-range dependence

Applicable to ultra-high-dimensional time series with sub-exponential growth in dimension

Abstract

For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance between the infinity norms of the estimation error of sample covariance matrix and the corresponding Gaussian approximation. The method utilizes martingale and m-dependent approximation and relies on constructing triadic blocks. We also establish a bootstrapping result with block sampling method, which preserves validity despite strong temporal dependence. Our results on covariance allow ultra-high-dimensional settings where the dimension of time series can grow sub-exponentially with sample size. Similar results can be built for precision matrix under low-dimensional settings. No assumption is required on the structure of covariance and precision matrices.