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In the first part of this paper we give an elementary proof of the fact thatif an infinite matrix $A$, which is invertible as a bounded operator on$\ell^2$, can be uniformly approximated by banded matrices then so can theinverse 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 termsof their band-width. In the second part we apply these results to covariancematrices $\Sigma$ of Gaussian processes and study mixing and beta mixing ofprocesses in terms of properties of $\Sigma$. Finally, we note someapplications of our results to statistics.

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