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Showing papers on "Stochastic process published in 1971"



Journal ArticleDOI
TL;DR: In this paper, the role of the linear-quadratic stochastic control problem in engineering design is reviewed in tutorial fashion, motivated by considering the control of a non-linear uncertain plant about a desired input-output response.
Abstract: The role of the linear-quadratic stochastic control problem in engineering design is reviewed in tutorial fashion. The design approach is motivated by considering the control of a non-linear uncertain plant about a desired input-output response. It is demonstrated how a design philosophy based on 1) deterministic perturbation control, 2) stochastic state estimation, and 3) linearized stochastic control leads to an overall closed-loop control system. The emphasis of the paper is on the philosophy of the design process, the modeling issue, and the formulation of the problem; the results are given for the sake of completeness, but no proofs are included. The systematic off-line nature of the design process is stressed throughout.

611 citations


Journal ArticleDOI
TL;DR: In this article, the authors present efficient and practical methods of simulating multivariate and multidimensional processes with specified cross-spectral density matrix, which can be expressed as the sum of cosine functions with random frequencies and random phase angles.
Abstract: Efficient and practical methods of simulating multivariate and multidimensional processes with specified cross‐spectral density are presented. When the cross‐spectral density matrix of an n‐variate process is specified, its component processes can be simulated as the sum of cosine functions with random frequencies and random phase angles. Typical examples of this type are the simulation, for the purpose of shaker test, of a multivariate process representing six components of the acceleration (due to, for example, a booster engine cutoff) measured at the base of a spacecraft and the simulation of horizontal and vertical components of earthquake acceleration. A homogeneous multidimensional process can also be simulated in terms of the sum of cosine functions with random frequencies and random phase angles. Examples of multidimensional processes considered here include the horizontal component f0(t,x) of the wind velocity perpendicular to the axis (x axis) of a slender structure, the vertical gust velocity field f0(x,y) frozen in space, and the boundary‐layer pressure field f0(x,y,t). Also, a convenient use of the present method of simulation in a class of nonlinear structural vibration analysis is described with a numerical example.

582 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived thehentsov-Billingsley type fluctuation inequalities for stochastic processes whose time parameter ranges over the $q$-dimensional unit cube and established weak convergence results for such processes.
Abstract: Chentsov-Billingsley type fluctuation inequalities for stochastic processes whose time parameter ranges over the $q$-dimensional unit cube are derived and used to establish weak convergence results for such processes.

517 citations


Book ChapterDOI
TL;DR: In this article, the authors present modeling in nonlinear random vibrations by Markov processes, and discuss the basic theory of stochastic processes and its applications and solution techniques, and the difficulties involved in modeling non-linear random vibrational effects.
Abstract: Publisher Summary Random vibration analysis of mechanical systems has become an important subject in recent years, principally because of advances in high speed flight. To design structures and equipment that will survive the randomly fluctuating loads caused by the flow of turbulent air or the efflux of jet or rocket engines, it has become necessary to develop a theory capable of analyzing the effect of such fluctuating loads on structures and equipment. Many of the techniques developed for the analysis of random excitation of nonlinear control systems are applicable to the analysis of nonlinear random vibrations, and conversely many of the techniques developed in the theory of nonlinear random vibrations are equally applicable to problems in communication theory and electronics. The chapter presents modeling in nonlinear random vibrations by Markov processes. The chief reason for adopting the idealized model of a system of differential equations excited by white noise is that the computations are much simpler in this case. One of the difficulties involved in modeling nonlinear random vibrations by Markov processes is that—one is restricted to quasi-linear systems. In the subsequent development of the theory, no distinction has been made between the physical nonlinearity and the mathematical model of that nonlinearity. Further, the chapter also discusses the basic theory of stochastic processes and its applications and solution techniques.

260 citations


Book
01 Nov 1971

231 citations


Journal ArticleDOI
TL;DR: In this article, the authors discuss proposals for testing the presence of phase relations and for extracting them quantitatively by means of numerical bispectrum analysis, and derive their statistical properties and compare their relative merits.
Abstract: Harmonically related peaks in the spectrum of a stationary stochastic process may indicate the presence or wave components that are not sine-shaped, i.e., whose Fourier expansions contain phase-locked higher order terms. But the spectrum itself suppresses phase relations, and more refined methods are needed to decide such questions. Moreover, phase relations might also exist outside of the peaks. We discuss proposals for testing the presence of phase relations and for extracting them quantitatively by means of numerical bispectrum analysis, and we derive their statistical properties and compare their relative merits. Applications of these methods to EEG signals will be presented.

206 citations


Journal ArticleDOI
Hiroshi Kunita1
TL;DR: In this article, it was shown that the invariant measure of the filtering process exists uniquely if and only if the stationary signal process (flow) is purely non-deterministic.

181 citations


Journal ArticleDOI
TL;DR: It is shown how the Karhunen-Loeve approach to the detection of a deterministic signal can be given a coordinate-free and geometric interpretation in a particular Hilbert space of functions that is uniquely determined by the covariance function of the additive Gaussian noise.
Abstract: First it is shown how the Karhunen-Loeve approach to the detection of a deterministic signal can be given a coordinate-free and geometric interpretation in a particular Hilbert space of functions that is uniquely determined by the covariance function of the additive Gaussian noise. This Hilbert space, which is called a reproducing-kernel Hilbert space (RKHS), has many special properties that appear to make it a natural space of functions to associate with a second-order random process. A mapping between the RKHS and the linear Hilbert space of random variables generated by the random process is studied in some detail. This mapping enables one to give a geometric treatment of the detection problem. The relations to the usual integral-equation approach to this problem are also discussed. Some of the special properties of the RKHS are developed and then used to study the singularity and stability of the detection problem and also to suggest simple means of approximating the detectability of the signal. The RKHS for several multidimensional and multivariable processes is presented; by going to the RKHS of functionals rather than functions it is also shown how generalized random processes, including white noise and stationary processes whose spectra grow at infinity, are treated.

143 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approxima.
Abstract: It is shown that there is a simple relation between master equation and random walk solutions. We assume that the random walker takes steps at random times, with the time between steps governed by a probability density ψ(Δt). Then, if the random walk transition probability matrix M and the master equation transition rate matrix A are related by A = (M − 1)/τ1, where τ1 is the first moment of Ψ(t) and thus the average time between steps, the solutions of the random walk and the master equation approach each other at long times and are essentially equal for times much larger than the maximum of (τn/n!)1/n, where τn is the nth moment of ψ(t). For a Poisson probability density ψ(t), the solutions are shown to be identical at all times. For the case where A ≠ (M − 1)/τ1, the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approxima...

133 citations


Journal ArticleDOI
TL;DR: In this article, the authors deal with stochastic control of linear systems in which the information available to distinct controllers is different, and they obtain necessary conditions for optimality of the controller structure parameters.
Abstract: This paper deals with the stochastic control of linear systems in which the information available to distinct controllers is different. Constraints on the control structure are imposed. Necessary conditions for optimality of the controller structure parameters are obtained. The results show that the separation theorem does not hold in this case.

Journal ArticleDOI
TL;DR: In this article, the first-excursion probability of a stationary narrow-band Gaussian process with mean zero has been studied in the framework of point process approach, series approximations derived from the theory of random points and approximation based on the maximum entropy principle have been developed.
Abstract: The first-excursion probability of a stationary narrow-band Gaussian process with mean zero has been studied. Within the framework of point process approach, series approximations derived from the theory of random points and approximations based on the maximum entropy principle have been developed. With the aid of numerical examples, merits of the approximations proposed previously as well as of those developed in this paper have been compared. The results indicate that the maximum entropy principle has not produced satisfactory approximations but the approximation based on nonapproaching random points is found to be the best among all the approximations proposed herein. A conclusion drawn from the present and the previous studies is that the point process approach produces a number of useful approximations for the first-excursion probability, particularly those based on the concepts of the Markov process, the clump-size, and the nonapproaching random points.

Journal ArticleDOI
TL;DR: In this article, the stability of a class of dynamical systems containing random parameters is investigated and an input-output stability concept is formulated for stochastic systems, and necessary and sufficient frequency domain conditions for stability are derived and it is shown that the uncertainty has a destabilizing effect.
Abstract: The stability of a class of dynamical systems containing random parameters is investigated. An input-output stability concept is formulated for stochastic systems. The specific class of systems considered consists of those feedback systems whose open loop consists of the cascade of a white noise multiplicative gain and a linear deterministic dynamical system. Necessary and sufficient frequency domain conditions for stability are derived and it is shown that the uncertainty has a destabilizing effect. The resulting stability conditions depend on the open-loop stability, the rms value of the stochastic gain element, and the effective bandwidth of the linear element.

Journal ArticleDOI
H. Ogura1
TL;DR: This paper deals with the periodic nonstationary process, the mean value and the correlation function of which are invariant under shift by a multiple of a certain period and are represented in terms of a matrix-valued spectral density that is hermitian and nonnegative definite.
Abstract: This paper deals with the periodic nonstationary process, the mean value and the correlation function of which are invariant under shift by a multiple of a certain period. The spectral representation is derived by making use of Loeve's harmonizability theorem on a second-order nonstationary process. The process is represented as a sum of infinite stationary processes among which covariances exist. Each stationary process has a nonoverlapping frequency band of equal width, the center of which corresponds to a harmonic of the fundamental frequency determined by the period. The correlation function, dependent on two points, is represented in terms of a matrix-valued spectral density that is hermitian and nonnegative definite. The representations in other possible forms are also given. Finally some properties, special processes, and examples produced by a certain stationary random sequence are discussed.

Journal ArticleDOI
TL;DR: In this article, a kinetic approach to fluctuations and correlations of stochastic processes depending on a continuous set of parameters y is presented, where particle densities np(y, t) refer to macroscopic densities in position space, or to microscopic quantities such as distributions in phase space, and occupancies of quantum states of which the labeling is continuous.
Abstract: A kinetic approach to fluctuations and correlations of stochastic processes depending on a continuous set of parameters y is presented. In particular, we consider particle densities np(y, t) which may refer to macroscopic densities in position space, or to microscopic quantities such as distributions in phase space, or occupancies of quantum states of which the labeling is continuous (as with Bloch states in solids). In a Markovian sense such processes are infinite dimensional. We describe the fluctuating particle densities in a Hilbert space: the analog of de Groot's a‐space for non‐spatial‐dependent variables. Mainly, we employ a Langevin description; i.e., we start from presumed phenomenological equations, amended with source densities ξp(y, t). A theorem is derived for the density‐density or two‐point covariance function (Λ theorem). In its general form, the theorem applies to the nonequilibrium steady state. It closely resembles the generalized g‐r theorem for finite‐dimensional processes. However, t...

Journal ArticleDOI
TL;DR: In this paper, a model of the pulse frequency modulation process in those neural systems where the neuron discharge is random is proposed, and the model is characterized by one property, namely input-invariance of the output random process after a time transformation, which greatly simplifies its analytical treatment, and gives a tool to determine experimentally whether the model describes the external behavior of a given neural system.

Journal ArticleDOI
TL;DR: In this paper, the authors considered a long earthquake sequence is considered to be a stationary stochastic process, and the stored elastic energy of deformation can be shown to be an independent variable in the usual backward equation.
Abstract: If a long earthquake sequence is considered to be a stationary stochastic process, the stored elastic energy of deformation can be shown to be an independent variable in the usual ‘backward’ equation. Three unknown probability functions are introduced: the probability that the stored energy of deformation is at a certain level; the probability that, if this energy is at a given level, an earthquake will occur; and the transition probability that, if the earthquake occurs, the final energy state will be at a certain level. It is assumed that the frequency-energy distribution is known. The equations can be solved, if the transition probability is assumed to be known; and they have been solved for the model in which the transition probability is a function of the energy released in the shock but is not otherwise dependent on the final energy state. In this case, the results can be used to describe the earthquake history for some time after a great shock, and possibly for times just before a great shock. The results have some features of inconsistency with observations.

Journal ArticleDOI
TL;DR: Optimal and suboptimal estimator algorithms, which can be computed recursively, are developed and digital computer simulations of the estimators are also presented and their performances evaluated.

Journal ArticleDOI
TL;DR: A broad class of random sampling schemes, for which the sampling intervals are dependent, is constructed and it is shown that this class is “alias free≓ relative to various families of spectra”.
Abstract: This paper deals with the problem of perfect reconstruction of the spectrum of a weakly stationary stochastic process x ( t ) from a set of random samples { x ( t n )}. A broad class of random sampling schemes, for which the sampling intervals are dependent, is constructed. It is shown that this class is “alias free≓ relative to various families of spectra. It is further shown that the alias free property of this class of sampling schemes is invariant under random deletion of samples.

Journal ArticleDOI
TL;DR: In this article, the authors assume that the paths of a stochastic process are right continuous and have left limits at each point, and they also assume that X(0) = 0.
Abstract: Let X = {X(t), t >= 0} be a stochastic process in R N with stationary, independent increments; suppose X is defined on some probability space (~, ~, P). We may (and will) assume that the paths of X are right continuous and have left limits at each point. We also assume throughout that X(0) = 0. If ~Pdu) is the (N-dimensional) characteristic function of X(t), then necessarily (see [4, 13, 16]) ~pt(u)= exp { ttp (u)}, where

Journal ArticleDOI
TL;DR: Theorem 5.1 of Billingsley as mentioned in this paper shows that weak convergence X n ⇒ X in D implies weak convergence S (X n ) ⇒ S ( X ) in D by virtue of the continuous mapping theorem.
Abstract: Let D = D [0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic process X in D, let the associated supremum process be S ( X ), where for any x ∊ D . It is easy to verify that S : D → D is continuous in any of Skorohod's (1956) topologies extended from D [0,1] to D [0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergence X n ⇒ X in D implies weak convergence S ( X n ) ⇒ S ( X ) in D by virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

Journal ArticleDOI
TL;DR: A result is presented which generalizes the paper of Watanabe (Trans. AMS 148, 233-248) and is extended also to the nonstationary process treated by Watanbe.
Abstract: : The paper presents a result which generalizes the paper of Watanabe (Trans. AMS 148, 233-248). The result is extended also to the nonstationary process treated by Watanabe.


Proceedings ArticleDOI
01 Dec 1971
TL;DR: A class of nonstationary environments with unknown but periodically changing probabilistic characteristics is considered and a two-level system of variable-structure stochastic automata is proposed to optimize the performance.
Abstract: A class of nonstationary environments with unknown but periodically changing probabilistic characteristics is considered. It is proposed to optimize the performance in an environment from this class by using a two-level system of variable-structure stochastic automata. The first level estimates the unknown period while the second level operates suitably in the environment for one cycle assuming that this estimate is the true period of the environment. The average output of the environment in this cycle is used as the input to the first level to determine the next estimate of the period. The optimal performance of this two-level system of automata in periodic random environments is demonstrated through computer simulations.

Journal ArticleDOI
TL;DR: For moments of normal random variables with nonzero means, this paper showed that higher order moments can be expressed in terms of second-order moments, which is the same as the second order moments of moments of a normal random variable with zero means.
Abstract: It is well known that higher order moments of normal random variables (RV) with zero means can be expressed by terms of second-order moments. In this correspondence an extension will be given for moments of normal RV with nonzero means.

Journal ArticleDOI
TL;DR: For weakly stationary stochastic processes taking values in a Hilbert space, spectral representation and Cramer decomposition are studied in this article, where necessary and sufficient spectral conditions for such stochastically processes to be purely non-deterministic are given in both discrete and continuous parameter cases.

Journal ArticleDOI
TL;DR: In this paper, the problem of wave motion in a stochastic medium is treated as an application of stochastically operator theory and as a generalization of papers I and II (and previous work by the author) to the case of partial differential equations and random fields without monochromaticity assumptions and closure approximations.
Abstract: The problem of wave motion in a stochastic medium is treated as an application of stochastic operator theory and as a generalization of Papers I and II (and previous work by the author) to the case of partial differential equations and random fields without monochromaticity assumptions and closure approximations. Connections to the theory of partial coherence are considered. The stochastic Green's function for the two‐point correlation of the solution process can be determined so the correlation can be obtained. Spectral spreading in a ``hot'' medium is easily demonstrable and can be calculated.

Journal ArticleDOI
TL;DR: In this article, a procedure for determining an optimal zero-memory regulator for a linear stochastic plant with an incomplete probabilistic description is given for situations in which the initial ambiguity is modeled probabilistically.
Abstract: A procedure is given for determining an optimal zero-memory regulator for a linear stochastic plant with an incomplete probabilistic description. For situations in which the initial ambiguity is modeled probabilistically, the equations characterizing the appropriate Bayes control are derived. If no prior distribution is available, a minimax controller is sought and an algorithm for obtaining it is described.