Stationary Gaussian Vector Processes Research Articles

Method of Moments for Exit Probabilities of Gaussian Vector Processes From a Large Region

Asymptotic behavior of the exit probability from homothetically unboundedly expanding region is evaluated for twice continuously differentiable stationary Gaussian vector processes. The evaluation ...

Piterbarg's max-discretization theorem for stationary vector Gaussian processes observed on different grids

In this paper, we derive Piterbarg's max-discretization theorem for two different grids considering centred stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum of Gaussian processes over and over a grid . In this paper, we extend the recent findings by considering additionally the maximum over another grid . We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choices of such grids by letting . As a by-product we find that the joint limiting distribution of the maximum over different grids, which we refer to as the Piterbarg distribution, is in the case of weakly dependent Gaussian processes a max-stable distribution.

A central limit theorem for non-linear functionals of stationary Gaussian vector processes

Let X n = ( X n 1, …, X n d ) be a stationary Gaussian vector process, such that the correlation matrix tends fast to 0. Suppose H is a d-dimensional function and define Z N(H) = ( N ) −1 ∑ j=1 N H(X j) . It is shown that Z N ( H) has a Gaussian limiting distribution.

Methods for Time‐Dependent Reliability and Sensitivity Analysis

A general and efficient method for reliability and sensitivity analysis is presented. The methods are for the analysis of components and systems in design situations where uncertainties are represented by a vector of random variables and a stationary Gaussian vector process. A formulation as a first‐passage problem for a vector process outcrossing a safe set is first applied for a fixed value of the random variable vector. This gives a conditional failure probability, and a fast integration technique based on a first‐ or second‐order reliability method is then applied to compute the unconditional failure probability. Extensive use of analytical gradient information is made in the iteration algorithms and in the calculation of sensitivity factors. A nested first‐order reliability method is proposed together with a method based on an integrated optimization for the two first‐order reliability analyses. For a preliminary analysis, a “crude” first‐order reliability method is described. The computation time is only a few times the computation time for a first‐order reliability analysis of a similar time‐independent problem. Software developed from existing special software for first‐order reliability analysis or existing general optimization software can be applied.

Asymptotic crossing rates for stationary Gaussian vector processes

For stationary differentiable Gaussian vector processes the expected number of crossings through a hypersurface is given by a surface integral. In general, this is difficult to calculate. In this paper asymptotic approximations for these surface integrals are derived.

Realization Theory for Multivariate Stationary Gaussian Processes

This paper collects in one place a comprehensive theory of stochastic realization for continuous-time stationary Gaussian vector processes which in various pieces has appeared in a number of our earlier papers. It begins with an abstract state space theory, based on the concept of splitting subspace. These results are then carried over to the spectral domain and described in terms of Hardy functions. Finally, differential-equations type stochastic realizations are constructed. The theory is coordinate-free, and it accommodates infinite-dimensional representations, minimality and other systems-theoretical concepts being defined by subspace inclusion rather than by dimension. We have strived for conceptual completeness rather than generality, and the same framework can be used for other types of stochastic realization problems.

Failure rates for general systems including structural components

At present, the methods of analysis of the reliability of man-machine-structure systems appear inconsistent and difficult, particularly if both the results from a classical and from the structural analysis have to be combined. It is shown that the unconditional failure rates in classical reliability theory have the same meaning as the outcrossing rates in structural reliability so that both approaches can be unified. Some practically important cases of systems, i.e. systems under the excitation of stationary Gaussian vector processes, are studied in more detail.

On the rate of convergence of the distribution of the periodogram matrix for gaussian vector processes

Let {(Xk(t))k=1, …, p}t=… -1,0,1, … be a stationary Gaussian vector process with Hermitian positive definite spectral density matrix f(λ). We assume, that for some αwith 0<α≦1 and for j, k = 1, …, ...

On a condition for minimality of Markovian splitting subspaces

Given a stationary Gaussian vector process, consider a Markovian splitting subspace X contained in the frame space which is either observable or constructible. Such an X will be called reduced . In this paper we show that a Markovian splitting subspace is minimal if and only if it is reduced. This was claimed in some earlier papers but there are nontrivial gaps in the proofs presented there. The proof is based on a lemma staling that all reduced X have quasi-equivalent structural functions. This property is also important in isomorphism theory for minimal splitting subspaces.

The Estimation of an Autocorrelation Parameter of a Gaussian Vector Process

Summary A procedure based on differences is used to yield an estimator of an unknown parameter occurring in the autocorrelation structure of a stationary Gaussian vector process. Various properties of the estimator including its existence, consistency and asymptotic normality are shown. Necessary and sufficient conditions are given for the estimator to be efficient.

On the Cross Periodogram of a Stationary Gaussian Vector Process

Bartlett [1] obtained an asymptotic relation connecting the periodogram of a linear process and the periodogram of the residuals with uniform spectrum in the case of one dimensional stationary time series, and using this relation he studied the asymptotic properties of the periodogram. In this paper we obtain some asymptotic relations between the co-periodogram and quadrature periodogram of the residuals in the case of a stationary Gaussian vector process. The covariances of the co-periodogram and quadrature periodogram have also been obtained. Two inequalities connecting the variance and the bandwidth of the cross spectral estimate have been derived.

Estimation of the Cross-Spectrum

Assuming that the wide sense stationary process being sampled has an absolutely continuous spectrum, Parzen [5] has shown the consistency of a general class of estimators for estimating the spectral density at a given frequency. He has shown that the class of estimators he considers, contains as a particular case estimators considered earlier by Grenander and Rosenblatt [2], Barlett [1], Tukey [7], [8], Daniell, and Lomnicki and Zaremba [3]. In this paper we extend Parzen's results to the case of a Stationary Gaussian Vector process whose spectrum is not necessarily absolutely continuous and show that this general class of estimators consistently estimate the co- and quadrature spectral densities at all those frequencies where they exist. It has been earlier shown by the author [4], using an entirely different approach from the one presented in this paper, that for a normal Stationary process whose spectrum besides the absolutely continuous part contains a step function with a finite number of saltuses, the weighted periodogram estimator, which is a particular case of the general class of estimators considered by Parzen [5], is still a consistent estimate of the spectral density at any point of continuity of the spectrum. Thus this paper also substantially generalizes the earlier result of the author, where he limits himself to a finite number of saltuses.