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.
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机译：在本文的第一部分中，我们给出以下事实的基本证明：如果无穷大的矩阵$ A $作为带界算子在$ \ ell ^ 2 $上可逆，则可以由带状矩阵统一逼近，那么， $ A $。我们给出了$ A ^ {-1} $的带近似值的明确公式，以及它们在带宽方面的准确性和收敛速度的界限。在第二部分中，我们将这些结果应用于高斯过程的协方差矩阵$ \ Sigma $，并根据$ \ Sigma $的性质研究过程的混合和Beta混合。最后，我们注意到我们的结果在统计学中的一些应用。
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