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
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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.
@article{TVP_2011_56_1_a4,
author = {P. Bickel and M. Lindner},
title = {Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {100--122},
publisher = {mathdoc},
volume = {56},
number = {1},
year = {2011},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_2011_56_1_a4/}
}
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%0 Journal Article %A P. Bickel %A M. Lindner %T Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics %J Teoriâ veroâtnostej i ee primeneniâ %D 2011 %P 100-122 %V 56 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/TVP_2011_56_1_a4/ %G en %F TVP_2011_56_1_a4
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/