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

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 ${\mathrm{\ell }}^2$, 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 $\mathrm{\Sigma }$ of Gaussian processes and study mixing and beta mixing of processes in terms of properties of $\mathrm{\Sigma }$. Finally, we note some applications of our results to statistics.