Geodesic
Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics
Teoriâ veroâtnostej i ee primeneniâ, Tome 56 (2011) no. 1, pp. 100-122 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. We finally use these results to prove that the so-called Wiener algebra is inverse closed. 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.
Keywords: infinite band-dominated matrices, Gaussian stochastic processes, mixing conditions, high dimensional statistical inference. 
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P. Bickel; M. Lindner. Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics. Teoriâ veroâtnostej i ee primeneniâ, Tome 56 (2011) no. 1, pp. 100-122. http://geodesic.mathdoc.fr/item/TVP_2011_56_1_a4/

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