Related papers: On the trace approximations of products of Toeplit…
We investigate error orders for integral limit approximations to traces of products of Toeplitz matrices generated by integrable functions on $[-\pi,\pi]$ having some singularities at the origin. Even though a sharp error order of the above…
The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szeg\"o, "Toeplitz forms and their applications". It has then been extensively studied in the…
We study the approximation of stationary processes by a simple class of purely deterministic signals. This has an analytic counterpart in the approximation of symmetric positive definite Toeplitz matrices by submatrices of finite rank. We…
We derive sharp approximation error bounds for inverse block Toeplitz matrices associated with multivariate long-memory stationary processes. The error bounds are evaluated for both column and row sums. These results are used to prove the…
When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for…
We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in $\mathcal{O}(n \log n)$ time with applications to fitting time series models. In particular, for a symmetric…
Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis and…
We obtain a sharp convergence rate for banded covariance matrix estimates of stationary processes. A precise order of magnitude is derived for spectral radius of sample covariance matrices. We also consider a thresholded covariance matrix…
We derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function…
In several applications, one must estimate a real-valued (symmetric) Toeplitz covariance matrix, typically shifted by the conjugated diagonal matrices of phase progression and phase "calibration" errors. Unlike the Hermitian Toeplitz…
This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or L\'evy-driven linear processes with memory, and are defined…
We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criteria for a…
We present some results concerning the almost sure behaviour of the operator norm or random Toeplitz matrices, including the law of large numbers for the norm, normalized by its expectation (in the i.i.d. case). As tools we present some…
We analyse the convergence of the ergodic formula for Toeplitz matrix-sequences generated by a symbol and we produce explicit bounds depending on the size of the matrix, the regularity of the symbol and the regularity of the test function.
Trace and extreme eigenvalues of a product of truncated Toeplitz matrices. The singular case. In a first theorem we give an asymptotic expansion of Tr (T_N (f_1) T_N^{-1}(f_2)) where f1 ({\theta}) = |1 - e^{i {\theta}} | ^{2{{\alpha}1}c1…
In this paper, we survey some recent results on statistical inference (parametric and nonparametric statistical estimation, hypotheses testing) about the spectrum of stationary models with tapered data, as well as, a question concerning…
Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that…
This study investigates tridiagonal near-Toeplitz matrices in which the Toeplitz part is strictly diagonally dominant. The focus is on determining the exact inverse of these matrices and establishing upper bounds for the infinite norms of…
In this paper, we provide explicit formulas for the exact inverses of the symmetric tridiagonal near-Toeplitz matrices characterized by weak diagonal dominance in the Toeplitz part. Furthermore, these findings extend to scenarios where the…
In this paper, we study products of asymmetric Toeplitz matrices, we give necessary and sufficient conditions for the product of two asymmetric Toeplitz matrices compatible sizes is asymmetric Toeplitz matrix. We also give some results…