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Necessary and sufficient conditions for AR vector processes to be stationary: Applications in information theory and in statistical signal processing
Abstract
As the correlation matrices of stationary vector processes are block Toeplitz, autoregressive (AR) vector processes are non-stationary. However, in the literature, an AR vector process of finite order is said to be “stationary” if it satisfies the so-called stationarity condition (i.e., if the spectral radius of the associated companion matrix is less than one). Since the term “stationary” is used for such an AR vector process, its correlation matrices should “somehow approach” the correlation matrices of a stationary vector process, but the meaning of “somehow approach” has not been mathematically established in the literature. In the present paper we give necessary and sufficient conditions for AR vector processes to be “stationary”. These conditions show in which sense the correlation matrices of an AR “stationary” vector process asymptotically behave like block Toeplitz matrices. Applications in information theory and in statistical signal processing of these necessary and sufficient conditions are also given.
... Finally, it should be noted that the assumptions required for Corollary 1 are more restrictive than the ones in Theorem 4 because taking ρ(Ψ q (B)) < 1 implies, by [10,Theorem 6], that det(B(ω)) = 0 for all ω ∈ R. ...
In the present article, the differential entropy rate and the rate distortion function (RDF) are computed for certain nonstationary real Gaussian autoregressive moving average (ARMA) vector sources.
Для определенных нестационарных источников вещественных гауссовских векторных процессов авторегрессии со скользящим средним (ARMA-процессов) вычисляются скорость создания дифференциальной энтропии и скорость как функция искажения.
In this paper, we study the asymptotic optimality of a low-complexity coding strategy for Gaussian vector sources. Specifically, we study the convergence speed of the rate of such a coding strategy when it is used to encode the most relevant vector sources, namely wide sense stationary (WSS), moving average (MA), and autoregressive (AR) vector sources. We also study how the coding strategy considered performs when it is used to encode perturbed versions of those relevant sources. More precisely, we give a sufficient condition for such perturbed versions so that the convergence speed of the rate remains unaltered.
In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numerical discretization of a differential equation (DE), the knowledge of the spectral distribution of the associated matrix has proved to be a useful information for designing/analyzing appropriate solvers---especially, preconditioned Krylov and multigrid solvers---for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material). The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices A_n arising from virtually any kind of numerical discretization of DEs. Indeed, when the mesh-fineness parameter n tends to infinity, these matrices A_n give rise to a sequence {A_n}_n, which often turns out to be a GLT sequence or one of its "relatives", i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar/vectorial DEs. This work is a review, refinement, extension, and systematic exposition of the theory of block GLT sequences. It also includes several emblematic applications of this theory in the context of DE discretizations.
The present monograph studies the asymptotic behaviour of
eigenvalues, products and functions of block Toeplitz matrices
generated by the Fourier coefficients of a continuous matrix-valued
function. This study is based on the concept of asymptotically equivalent
sequences of non-square matrices. The asymptotic results on
block Toeplitz matrices obtained are applied to vector asymptotically
wide sense stationary processes. Therefore, this monograph is a generalization
to block Toeplitz matrices of the Gray monograph entitled
“Toeplitz and circulant matrices: A review”, which was published in
the second volume of Foundations and Trends in Communications
and Information Theory, and which is the simplest and most famous
introduction to the asymptotic theory on Toeplitz matrices.
The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of "finite section" Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hopes of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes. Acknowledgements The author gratefully acknowledges the assistance of Ronald M. Aarts of the Philips Research Labs in correcting many typos and errors in the 1993 revision, Liu Mingyu in pointing out errors corrected in the 1998 revision, Paolo Tilli of the Scuola Normale Superiore of Pisa for pointing out an incorrect corollary and providing the correction, and to David Neuho# of the University of Michigan for pointing out several typographical errors and some confusing notation. For corrections, comments, and improvements to the 2001 revision thanks are due to William Trench, John Dattorro, and Young Han-Kim. In particular, Trench brought the Wielandt-Ho#man theorem and its use to prove strengthened results to my attention. Section 2.4 largely follows his suggestions, although I take the blame for any introduced errors. Contents 1
The Pisarenko method estimates the power spectral density (PSD) of Gaussian wide sense stationary (WSS) 1-dimensional (scalar) processes when the PSD is Lipschitz. In this paper we modify the Pisarenko method to estimate the generating function of a sequence of block Toeplitz matrices from another sequence of matrices when both sequences are asymptotically equivalent. This modified version of the Pisarenko method allows us to estimate the PSD of any asymptotically WSS (AWSS) multidimensional (vector) process even if the process is not Gaussian and even if the PSD is continuous, but not Lipschitz.
The estimation of spectra of random stationary processes is an important part of the statistics of random processes. There are several books on spectral analysis, e.g. Blackman & Tukey, Hannan, and Jenkins & Watts. As a rule, spectral estimators are quadratic functions of the realizations. Recently Capon suggested a new method for estimation of spectra of random fields, in which a non-quadratic function of the realization is used: he considered a homogeneous random field ξ(t, x1, x2), i.e. one which is stationary in time and space and a random function of the time and space co-ordinates t, x1, x2. For the sake of expository convenience we shall consider ordinary stationary processes of time only, ξ(t); the generalization of our results to the case of random fields is easy.
Comparison of the conventional spectral estimator and the ‘high-resolution’ estimator for an artificial example showed that the latter has less smoothing effect on the true spectrum (Capon). This was later confirmed by examples using real data (Capon). However, it was not clear whether for a finite realization the high-resolution estimator distorted the true spectrum, i.e. whether it behaved for example like a conventional estimator raised to some power.
In the present paper we introduce and study a new class of spectral estimators which are generally non-linear and non-quadratic functionals of the realizations. These estimators include the conventional and high-resolution ones, for which we shall give the approximate distributions. We derive under rather general conditions the limiting distribution of the new class of estimators, and illustrate them with several examples. As a matter of fact, these new estimators are weighted means of the eigenvalues of the covariance matrix, e.g. the arithmetic mean, geometric mean, and so on.
Singular values and eigenvalues of non-Hermitian block Toeplitz matrices
- Tilli