Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics

TL;DR

This paper proves that invertible infinite matrices approximable by banded matrices have their inverses also approximable by banded matrices, and applies these results to covariance matrices in Gaussian processes and statistical contexts.

Contribution

It provides explicit formulas and bounds for banded approximations of matrix inverses, extending the theory to covariance matrices and statistical applications.

Findings

Inverse of banded approximable matrices can be similarly approximated by banded matrices.

Explicit formulas for banded inverse approximations are derived.

Applications to Gaussian process covariance matrices and statistical methods.

Abstract

In the first part of this paper we give an elementary proof of the fact that if an infinite matrix $A$, which is invertible as a bounded operator on $\ell^2$, can be uniformly approximated by banded matrices then so can the inverse of $A$. We give explicit formulas for the banded approximations of $A^{-1}$ as well as bounds on their accuracy and speed of convergence in terms of their band-width. In the second part we apply these results to covariance matrices $\Sigma$ of Gaussian processes and study mixing and beta mixing of processes in terms of properties of $\Sigma$. Finally, we note some applications of our results to statistics.