scholarly journals The prediction theory of multivariate stochastic processes: I. The regularity condition

1957 ◽  
Vol 98 (0) ◽  
pp. 111-150 ◽  
Author(s):  
N. Wiener ◽  
P. Masani
Keyword(s):  
Stochastic Processes ◽  
Regularity Condition ◽  
Prediction Theory

scholarly journals Non-Linear prediction problems for Ornstein-Uhlenbeck process

1983 ◽  
Vol 91 ◽  
pp. 173-184 ◽  
Author(s):  
Sheu-San Lee
Keyword(s):  
Stochastic Processes ◽  
Linear Functional ◽  
Linear Prediction ◽  
Basic Process ◽  
Prediction Theory ◽  
Uhlenbeck Process ◽  
Linear Predictor ◽  
Ornstein Uhlenbeck Process ◽  
Non Linear ◽  
Prediction Problems

We shall discuss in this paper some problems in non-linear prediction theory. An Ornstein-Uhlenbeck process {U(t)} is taken to be a basic process, and we shall deal with stochastic processes X(t) that are transformed by functions f satisfying certain condition. Actually, observed processes are expressed in the form X(t) = f(U(t)). Our main problem is to obtain the best non-linear predictor X̂(t, τ) for X(t + τ), τ > 0, assuming that X(s), s ≤t, are observed. The predictor is therefore a non-linear functional of the values X(s), s ≤ t.

Approximating integrals of stochastic processes: extensions

1998 ◽  
Vol 35 (04) ◽  
pp. 843-855
Author(s):  
Karim Benhenni
Keyword(s):  
Stochastic Processes ◽  
Mean Square Error ◽  
Regularity Condition ◽  
Approximation Error ◽  
Mean Square ◽  
Asymptotic Results ◽  
Linear Estimators ◽  
Regular Sampling ◽  
The Mean ◽  
Maclaurin Formula

We consider the problem of predicting integrals of second order processes whose covariances satisfy some Hölder regularity condition of order α > 0. When α is an odd integer, linear estimators based on regular sampling designs were constructed and asymptotic results for the approximation error were derived. We extend this result to any α > 0. When 2K < α ≤ 2K + 2, K a non-negative integer, we use an appropriate predictor based on the Euler-MacLaurin formula of order K with regular sampling designs. We give the corresponding result for the mean square error.

scholarly journals Superoscillations and Analytic Extension in Schur Analysis

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Daniel Alpay ◽  
Fabrizio Colombo ◽  
Irene Sabadini
Keyword(s):  
Stochastic Processes ◽  
Positive Definite ◽  
Signal Theory ◽  
Positive Definite Functions ◽  
Analytic Extension ◽  
Prediction Theory ◽  
Stationary Stochastic Processes

AbstractWe give applications of the theory of superoscillations to various questions, namely extension of positive definite functions, interpolation of polynomials and also of R-functions; we also discuss possible applications to signal theory and prediction theory of stationary stochastic processes. In all cases, we give a constructive procedure, by way of a limiting process, to get the required results.

Prediction Theory of Periodically Correlated Stochastic Processes

2015 ◽  
Author(s):  
Abolghassem Miamee ◽  
Andrzej Makagon
Keyword(s):  
Stochastic Processes ◽  
Prediction Theory

Approximating integrals of stochastic processes: extensions

1998 ◽  
Vol 35 (4) ◽  
pp. 843-855 ◽  
Author(s):  
Karim Benhenni
Keyword(s):  
Stochastic Processes ◽  
Mean Square Error ◽  
Regularity Condition ◽  
Approximation Error ◽  
Mean Square ◽  
Asymptotic Results ◽  
Linear Estimators ◽  
Regular Sampling ◽  
The Mean ◽  
Maclaurin Formula

We consider the problem of predicting integrals of second order processes whose covariances satisfy some Hölder regularity condition of order α > 0. When α is an odd integer, linear estimators based on regular sampling designs were constructed and asymptotic results for the approximation error were derived. We extend this result to any α > 0. When 2K < α ≤ 2K + 2, K a non-negative integer, we use an appropriate predictor based on the Euler-MacLaurin formula of order K with regular sampling designs. We give the corresponding result for the mean square error.

Remarks on the paper ‘Regenerative stochastic processes’

1960 ◽  
Vol 256 (1287) ◽  
pp. 496-501 ◽  
Keyword(s):  
Stochastic Processes ◽  
Regularity Condition ◽  
Step Function ◽  
Time Parameter ◽  
Regularity Conditions

The proof of a theorem in the above-mentioned paper (Smith, W. L. 1955 Proc. Roy. Soc . A, 232, 6) assumes more than it should and a new proof is given. For certain (unusual) kinds of Semi-Markor process the definition given in the aforementioned paper is inadequate to define the process for all values of the time parameter, and this weakness is removed. Furthermore, a ‘regularity’ condition for an S. M.- process to have step-function realizations (almost certainly) was given incorrectly; this error is now corrected, and certain other ‘regularity’ conditions are discussed.

Stochastic Processes in Demography and Applications

1992 ◽  
Vol 46 (1) ◽  
pp. 172-173
Author(s):  
S. Mitra
Keyword(s):  
Stochastic Processes
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