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

This article is cited in 10 scientific papers (total in 10 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.

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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Received: 28.02.2010
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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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    Citing articles on Google Scholar: Russian citations, English citations
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    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  zmath  isi
    2. Shao Meiyue, “On the finite section method for computing exponentials of doubly-infinite skew-Hermitian matrices”, Linear Algebra Appl., 451 (2014), 65–96  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  isi  scopus
    6. Cheng G., Zhang Zh., Zhang B., “Test For Bandedness of High-Dimensional Precision Matrices”, J. Nonparametr. Stat., 29:4 (2017), 884–902  crossref  mathscinet  zmath  isi
    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  crossref  isi
    8. Tong X.T., “Performance Analysis of Local Ensemble Kalman Filter”, J. Nonlinear Sci., 28:4 (2018), 1397–1442  crossref  mathscinet  isi
    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  crossref  mathscinet  isi  scopus
    10. Bien J., “Graph-Guided Banding of the Covariance Matrix”, J. Am. Stat. Assoc., 114:526 (2019), 782–792  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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