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 Teor. Veroyatnost. i Primenen., 2011, Volume 56, Issue 1, Pages 100–122 (Mi tvp4325)

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.

DOI: https://doi.org/10.4213/tvp4325

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English version:
Theory of Probability and its Applications, 2012, 56:1, 1–20

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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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp4325
• https://doi.org/10.4213/tvp4325
• http://mi.mathnet.ru/eng/tvp/v56/i1/p100

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bickel P.J., Levina E., Rothman A.J., Zhu J., “Minimax estimation of large covariance matrices under $l(1)$-norm comment”, Statist. Sinica, 22:4 (2012), 1367–1370
2. Shao Meiyue, “On the finite section method for computing exponentials of doubly-infinite skew-Hermitian matrices”, Linear Algebra Appl., 451 (2014), 65–96
3. Chen X., Wang Q., Wang X., “Truncation Approximations and Spectral Invariant Subalgebras in Uniform Roe Algebras of Discrete Groups”, J. Fourier Anal. Appl., 21:3 (2015), 555–574
4. Kurbatov V.G., Kuznetsova V.I., “Inverse-Closedness of the Set of Integral Operators With l-1-Continuously Varying Kernels”, J. Math. Anal. Appl., 436:1 (2016), 322–338
5. Wijewardhana U.L., Codreanu M., “A Bayesian Approach for Online Recovery of Streaming Signals from Compressive Measurements”, IEEE Trans. Signal Process., 65:1 (2017), 184–199
6. Cheng G., Zhang Zh., Zhang B., “Test For Bandedness of High-Dimensional Precision Matrices”, J. Nonparametr. Stat., 29:4 (2017), 884–902
7. Engel J., Buydens L., Blanchet L., “An Overview of Large-Dimensional Covariance and Precision Matrix Estimators With Applications in Chemometrics”, J. Chemometr., 31:4, SI (2017), e2880
8. Tong X.T., “Performance Analysis of Local Ensemble Kalman Filter”, J. Nonlinear Sci., 28:4 (2018), 1397–1442
9. Morzfeld M., Tong X.T., Marzouk Y.M., “Localization For Mcmc: Sampling High-Dimensional Posterior Distributions With Local Structure”, J. Comput. Phys., 380 (2019), 1–28
10. Bien J., “Graph-Guided Banding of the Covariance Matrix”, J. Am. Stat. Assoc., 114:526 (2019), 782–792
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