Related papers: Line Spectrum Representation for Vector Processes …
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…
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…
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…
Let $T_N$ denote an $N\times N$ Toeplitz matrix with finite, $N$ independent symbol ${\bf a}$. For $E_N$ a noise matrix satisfying mild assumptions (ensuring, in particular, that $N^{-1/2}\|E_N\|_{{\rm HS}}\to_{N\to\infty} 0$ at a…
Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite…
We consider banded block Toeplitz matrices $T_n$ with $n$ block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of $T_n$ for $n\to\infty$ weakly converges to one component of…
We provide an abstract framework for singular one-dimensional Schroedinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to…
In this article, we derive concentration inequalities for the spectral norm of two classical sample estimators of large dimensional Toeplitz covariance matrices, demonstrating in particular their asymptotic almost sure consistence. The…
A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form $A=T(a)+E$ where $T(a)$ is the Toeplitz matrix with entries $(T(a))_{i,j}=a_{j-i}$, for $a_{j-i}\in\mathbb C$, $i,j\ge 1$, while $E$ is a matrix representing a compact…
This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by…
The line spectral estimation problem consists in recovering the frequencies of a complex valued time signal that is assumed to be sparse in the spectral domain from its discrete observations. Unlike the gridding required by the classical…
We consider the eigenvalues of a fixed, non-normal matrix subject to a small additive perturbation. In particular, we consider the case when the fixed matrix is a banded Toeplitz matrix, where the bandwidth is allowed to grow slowly with…
This paper is concerned with the spectral properties of matrices associated with linear filters for the estimation of the underlying trend of a time series. The interest lies in the fact that the eigenvectors can be interpreted as the…
"Toeplitzification" or "redundancy (spatial) averaging", the well-known routine for deriving the Toeplitz covariance matrix estimate from the standard sample covariance matrix, recently regained new attention due to the important Random…
Gridless direction-of-arrival (DOA) estimation with multiple frequencies can be applied in acoustics source localization problems. We formulate this as an atomic norm minimization (ANM) problem and derive an equivalent regularization-free…
A matrix representation of the evolution operator associated with a nonlinear stochastic flow with additive noise is used to compute its spectrum. In the weak noise limit a perturbative expansion for the spectrum is formulated in terms of…
The singular value and spectral distribution of Toeplitz matrix sequences with Lebesgue integrable generating functions is well studied. Early results were provided in the classical Szeg{\H{o}} theorem and the Avram-Parter theorem, in which…
Taylor's formula holds significant importance in function representation, such as solving differential difference equations, ordinary differential equations, partial differential equations, and further promotes applications in visual…
We consider a data matrix $X:=C_N^{1/2}ZR_M^{1/2}$ from a multivariate stationary process with a separable covariance function, where $C_N$ is a $N\times N$ positive semi-definite matrix, $Z$ a $N\times M$ random matrix of uncorrelated…
Pseudospectral analysis serves as a powerful tool in matrix computation and the study of both linear and nonlinear dynamical systems. Among various numerical strategies, random sampling, especially in the form of rank-$1$ perturbations,…