NDLI: Approximating the Inverse of Banded Matrices by Banded Matrices with Applications to Probability and Statistics

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Bickel, P. Lindner, M.
Copyright Year	2012
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 bandwidth. We then use these results to prove that the so-called Wiener algebra is inverse closed. In the second part of the paper 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.
Starting Page	1
Ending Page	20
Page Count	20
File Format	PDF
ISSN	0040585X
DOI	10.1137/S0040585X97985224
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	1
Volume Number	56
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-02-15
Access Restriction	Subscribed
Subject Keyword	high dimensional statistical inference Gaussian stochastic processes infinite band-dominated matrices mixing conditions
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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Approximating the Inverse of Banded Matrices by Banded Matrices with Applications to Probability and Statistics