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Theory Probab. Appl. 56, pp. 1-20 (20 pages)
Approximating the Inverse of Banded Matrices by Banded Matrices with Applications to Probability and Statistics
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Add to FacebookIn 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 bandwidth. We then use these results to prove that the so-called Wiener algebra is inverse closed. In the second part of the paper 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.
© 2012 Society for Industrial and Applied Mathematics
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Received February 28, 2010
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