On random processes that are almost strict sense stationary

1976 ◽  
Vol 8 (4) ◽  
pp. 820-830 ◽  
Author(s):  
Dag Tj⊘stheim
Keyword(s):  
Stationary Process ◽  
Random Processes ◽  
Stationary Processes ◽  
Strict Sense ◽  
Wide Sense ◽  
Original Process ◽  
Extended Class ◽  
Gaussian Case

An extension of the class of strict sense stationary processes is studied. The extended class represents the strict sense analogy of an extension of wide sense stationary processes considered in an earlier paper [9]. The relations between the various types of processes defined are investigated in the general and in the Gaussian case, and some examples are given. It is shown that associated with a process belonging to the extended class there is a strict sense stationary process. The associated strict sense stationary process is unique iff the original process is ergodic.

On random processes that are almost strict sense stationary

1976 ◽  
Vol 8 (04) ◽  
pp. 820-830
Author(s):  
Dag Tj⊘stheim
Keyword(s):  
Stationary Process ◽  
Random Processes ◽  
Stationary Processes ◽  
Strict Sense ◽  
Wide Sense ◽  
Original Process ◽  
Extended Class ◽  
Gaussian Case

An extension of the class of strict sense stationary processes is studied. The extended class represents the strict sense analogy of an extension of wide sense stationary processes considered in an earlier paper [9]. The relations between the various types of processes defined are investigated in the general and in the Gaussian case, and some examples are given. It is shown that associated with a process belonging to the extended class there is a strict sense stationary process. The associated strict sense stationary process is unique iff the original process is ergodic.

Spectral generating operators for non-stationary processes

1976 ◽  
Vol 8 (4) ◽  
pp. 831-846 ◽  
Author(s):  
D. Tj⊘stheim
Keyword(s):  
Spectral Representation ◽  
Stationary Process ◽  
Adjoint Operator ◽  
Stationary Processes ◽  
Wide Sense ◽  
Linear Transformations ◽  
Generating Operators ◽  
The Time Domain ◽  
Image Position ◽  
Stationary Random Processes

A new method for obtaining spectral-like representations for a large class of non-stationary random processes is formulated. For a wide sense stationary process X(t) in continuous-time the spectral representation is generated by a self-adjoint operator H such that X(t)= eiHt X(0). Extending certain recently established operator identities for wide sense stationary processes, it is shown that similar operators exist for classes of non-stationary processes. The representation generated by such an operator has the form and it shares some of the properties of the wide sense stationary spectral representation: it is dual in a precisely defined sense to the time domain representation of X(t). There exists a class L of linear transformations of X(t) such that for G ∈ L for some function g determined by G.

Spectral generating operators for non-stationary processes

1976 ◽  
Vol 8 (04) ◽  
pp. 831-846 ◽  
Author(s):  
D. Tj⊘stheim
Keyword(s):  
Spectral Representation ◽  
Stationary Process ◽  
Adjoint Operator ◽  
Stationary Processes ◽  
Wide Sense ◽  
Linear Transformations ◽  
Generating Operators ◽  
The Time Domain ◽  
Image Position ◽  
Stationary Random Processes

A new method for obtaining spectral-like representations for a large class of non-stationary random processes is formulated. For a wide sense stationary process X(t) in continuous-time the spectral representation is generated by a self-adjoint operator H such that X(t)= eiHt X(0). Extending certain recently established operator identities for wide sense stationary processes, it is shown that similar operators exist for classes of non-stationary processes. The representation generated by such an operator has the form and it shares some of the properties of the wide sense stationary spectral representation: it is dual in a precisely defined sense to the time domain representation of X(t). There exists a class L of linear transformations of X(t) such that for G ∈ L for some function g determined by G.

scholarly journals Using Variability Related to Families of Spectral Estimators for Mixed Random Processes

2001 ◽  
Vol 123 (4) ◽  
pp. 572-584
Author(s):  
Li Wen ◽  
Changxue Wang ◽  
Peter Sherman
Keyword(s):  
Point Spectrum ◽  
Random Processes ◽  
Stationary Processes ◽  
Wide Sense ◽  
Dirac Delta ◽  
Harmonic Content ◽  
Spectral Estimators ◽  
Delta Functions ◽  
Spectral Estimator

Traditionally, characterization of spectral information for wide sense stationary processes has been addressed by identifying a single best spectral estimator from a given family. If one were to observe significant variability in neighboring spectral estimators then the level of confidence in the chosen estimator would naturally be lessened. Such variability naturally occurs in the case of a mixed random process, since the influence of the point spectrum in a spectral density characterization arises in the form of approximations of Dirac delta functions. In this work we investigate the nature of the variability of the point spectrum related to three families of spectral estimators: Fourier transform of the truncated unbiased correlation estimator, the truncated periodogram, and the autoregressive estimator. We show that tones are a significant source of bias and variability. This is done in the context of Dirichlet and Fejer kernels, and with respect to order rates. We offer some expressions for estimating statistical and arithmetic variability. Finally, we include an example concerning helicopter vibration. These results are especially pertinent to mechanical systems settings wherein harmonic content is prevalent.

A unique representation theorem for the conditional expectation of stationary processes and application to dynamic estimation problems

1997 ◽  
Vol 34 (2) ◽  
pp. 372-380
Author(s):  
Marco Campi
Keyword(s):  
Representation Theorem ◽  
Conditional Expectation ◽  
Stationary Process ◽  
Universal Function ◽  
Statistical Information ◽  
Stationary Processes ◽  
Strict Sense ◽  
Stationary Stochastic Processes ◽  
Estimation Problems ◽  
Dynamic Estimation

In this paper, multivariate strict sense stationary stochastic processes are considered. It is shown that there exists a universal function by means of which the conditional expectation of any stationary process with respect to its past can be represented. This requires no ergodicity assumptions. The important implications of this result in the evaluation of the achievable performance in certain dynamic estimation problems with incomplete statistical information are also discussed.

A unique representation theorem for the conditional expectation of stationary processes and application to dynamic estimation problems

1997 ◽  
Vol 34 (02) ◽  
pp. 372-380
Author(s):  
Marco Campi
Keyword(s):  
Representation Theorem ◽  
Conditional Expectation ◽  
Stationary Process ◽  
Universal Function ◽  
Statistical Information ◽  
Stationary Processes ◽  
Strict Sense ◽  
Stationary Stochastic Processes ◽  
Estimation Problems ◽  
Dynamic Estimation

In this paper, multivariate strict sense stationary stochastic processes are considered. It is shown that there exists a universal function by means of which the conditional expectation of any stationary process with respect to its past can be represented. This requires no ergodicity assumptions. The important implications of this result in the evaluation of the achievable performance in certain dynamic estimation problems with incomplete statistical information are also discussed.

Alternating projections and interpolation of stationary processes

1992 ◽  
Vol 29 (4) ◽  
pp. 921-931 ◽  
Author(s):  
Mohsen Pourahmadi
Keyword(s):  
Missing Values ◽  
Stationary Process ◽  
Dimensional Space ◽  
Stationary Processes ◽  
Finite Dimensional Space ◽  
Alternating Projections ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Explicit Formulae ◽  
Linear Interpolator

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

1991 ◽  
Vol 28 (01) ◽  
pp. 17-32 ◽  
Author(s):  
O. V. Seleznjev
Keyword(s):  
Gaussian Processes ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Processes ◽  
Trigonometric Polynomials ◽  
Stationary Processes ◽  
Periodic Processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

A note on the asymptotic eigenvalues and eigenvectors of the dispersion matrix of a second-order Stationary Process on a d-dimensional Lattice

1986 ◽  
Vol 23 (02) ◽  
pp. 529-535 ◽  
Author(s):  
R. J. Martin
Keyword(s):  
Time Series ◽  
Stationary Process ◽  
Second Order ◽  
Stationary Time Series ◽  
Stationary Processes ◽  
Dispersion Matrix ◽  
Eigenvalues And Eigenvectors ◽  
Dimensional Lattice ◽  
Stationary Time

A sufficiently large finite second-order stationary time series process on a line has approximately the same eigenvalues and eigenvectors of its dispersion matrix as its counterpart on a circle. It is shown here that this result can be extended to second-order stationary processes on a d-dimensional lattice.

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