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This article is cited in 17 scientific papers (total in 17 papers)
Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics
P. Bickela, M. Lindnerb a Department of Statistics, University of California, Berkeley
b Technische Universität Chemnitz, Fakultät für Mathematik
Abstract:
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.
Received: 28.02.2010
Citation:
P. Bickel, M. Lindner, “Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics”, Teor. Veroyatnost. i Primenen., 56:1 (2011), 100–122; Theory Probab. Appl., 56:1 (2012), 1–20
Linking options:
https://www.mathnet.ru/eng/tvp4325https://doi.org/10.4213/tvp4325 https://www.mathnet.ru/eng/tvp/v56/i1/p100
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Abstract page: | 973 | Full-text PDF : | 185 | References: | 57 |
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