Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics

2010/02/24 by Peter J. Bickel, Bickel, Peter J., Marko Lindner +1 · 1 citation
Computer Science · Mathematics · #Matrix Theory and Algorithms #Point processes and geometric inequalities #Random Matrices and Applications #math.FA #math.ST #msc:47B36 #msc:47L10 #msc:60G15 #msc:62H25 #msc:62M10 #msc:62M20 #stat.TH

paper · pdf · doi:10.48550/arxiv.1002.4545

arxiv created 2010/02/24 · arxiv updated 2010/02/26

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 ℓ², 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. In the second part we apply these results to covariance matrices Σ of Gaussian processes and study mixing and beta mixing of processes in terms of properties of Σ. Finally, we note some applications of our results to statistics.

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