Covariance matrix estimation for stationary time series

TL;DR

This paper establishes sharp convergence rates for banded and thresholded covariance matrix estimators of stationary processes, utilizing Toeplitz structure and spectral density relations to improve estimation accuracy.

Contribution

It introduces a novel approach combining Toeplitz ideas and spectral analysis to derive convergence rates and large deviation results for covariance matrix estimation in stationary time series.

Findings

Derived sharp convergence rates for banded covariance estimators.

Established a large deviation principle for quadratic forms of stationary processes.

Provided spectral density-based bounds for eigenvalues of covariance matrices.

Abstract

We obtain a sharp convergence rate for banded covariance matrix estimates of stationary processes. A precise order of magnitude is derived for spectral radius of sample covariance matrices. We also consider a thresholded covariance matrix estimator that can better characterize sparsity if the true covariance matrix is sparse. As our main tool, we implement Toeplitz [Math. Ann. 70 (1911) 351-376] idea and relate eigenvalues of covariance matrices to the spectral densities or Fourier transforms of the covariances. We develop a large deviation result for quadratic forms of stationary processes using m-dependence approximation, under the framework of causal representation and physical dependence measures.