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.

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.

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.

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.

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.

scholarly journals On the central limit theorem and iterated logarithm law for stationary processes

1975 ◽  
Vol 12 (1) ◽  
pp. 1-8 ◽  
Author(s):  
C.C. Heyde
Keyword(s):  
Central Limit ◽  
Stationary Process ◽  
Stationary Processes ◽  
Martingale Differences ◽  
Stationary Increments ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Functional Forms ◽  
Iterated Logarithm Law

It has recently emerged that a convenient way to establish central limit and iterated logarithm results for processes with stationary increments is to use approximating martingales with stationary increments. Functional forms of the limit results can be obtained via a representation for the increments of the stationary process in terms of stationary martingale differences plus other terms whose sum telescopes and disappears under suitable norming. Results based on the most general form of such a representation are here obtained.

Non-Stationary Processes and Spectrum

1968 ◽  
Vol 20 ◽  
pp. 1203-1206 ◽  
Author(s):  
K. Nagabhushanam ◽  
C. S. K. Bhagavan
Keyword(s):  
Spectral Density ◽  
Density Function ◽  
Stationary Process ◽  
Spectral Density Function ◽  
Stationary Processes ◽  
Gaussian Stationary Process ◽  
Discrete Parameter ◽  
Image Position

In 1964, L. J. Herbst (3) introduced the generalized spectral density Function1for a non-stationary process {X(t)} denned by1where {η(t)} is a real Gaussian stationary process of discrete parameter and independent variates, the (a;)'s and (σj)'s being constants, the latter, which are ordered in time, having their moduli less than a positive number M.

scholarly journals On a Necessary Condition for the Sample Path Continuity of Weakly Stationary Processes

1970 ◽  
Vol 38 ◽  
pp. 103-111 ◽  
Author(s):  
Izumi Kubo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Spectral Distribution ◽  
Covariance Function ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Sample Path ◽  
Necessary Condition ◽  
Stationary Processes ◽  
Spectral Distribution Function

We shall discuss the sample path continuity of a stationary process assuming that the spectral distribution function F(λ) is given. Many kinds of sufficient conditions have been given in terms of the covariance function or the asymptotic behavior of the spectral distribution function.

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.

Representation of Strongly Stationary Stochastic Processes

1993 ◽  
Vol 60 (3) ◽  
pp. 689-694 ◽  
Author(s):  
M. Di Paola
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Linear Systems ◽  
Stationary Processes ◽  
Orthogonality Conditions ◽  
Probabilistic Characterization ◽  
Stationary Stochastic Processes ◽  
Fixed Order ◽  
Strongly Stationary

A generalization of the orthogonality conditions for a stochastic process to represent strongly stationary processes up to a fixed order is presented. The particular case of non-normal delta correlated processes, and the probabilistic characterization of linear systems subjected to strongly stationary stochastic processes are also discussed.

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