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Showing papers on "Stochastic process published in 1973"


Journal ArticleDOI
TL;DR: In this article, two stochastic optimal control problems are solved whose performance criteria are the expected values of exponential functions of quadratic forms, and the optimal controller is linear in both cases but depends upon the covariance matrix of the additive process noise so that the certainty equivalence principle does not hold.
Abstract: Two stochastic optimal control problems are solved whose performance criteria are the expected values of exponential functions of quadratic forms. The optimal controller is linear in both cases but depends upon the covariance matrix of the additive process noise so that the certainty equivalence principle does not hold. The controllers are shown to be equivalent to those obtained by solving a cooperative and a noncooperative quadratic (differential) game, and this leads to some interesting interpretations and observations. Finally, some stability properties of the asymptotic controllers are discussed.

692 citations


Journal ArticleDOI
TL;DR: In this article, the authors introduce Probability One-Dimension Random Variables Functions of One Random Variable and Expectation Joint Probability Distributions Some Important Discrete Distributions some Important Continuous Distributions The Normal Distribution Random Samples and Sampling Distributions Parameter Estimation Tests of Hypotheses Design and Analysis of Single Factor Experiments: The Analysis of Variance Design of Experiments with Several Factors Simple Linear Regression and Correlation Multiple Regression Nonparametric Statistics Statistical Quality Control and Reliability Engineering Stochastic Processes and Queueing Statistical Decision Theory References
Abstract: Introduction and Data Description An Introduction to Probability One-Dimension Random Variables Functions of One Random Variable and Expectation Joint Probability Distributions Some Important Discrete Distributions Some Important Continuous Distributions The Normal Distribution Random Samples and Sampling Distributions Parameter Estimation Tests of Hypotheses Design and Analysis of Single-Factor Experiments: The Analysis of Variance Design of Experiments with Several Factors Simple Linear Regression and Correlation Multiple Regression Nonparametric Statistics Statistical Quality Control and Reliability Engineering Stochastic Processes and Queueing Statistical Decision Theory References Appendix Answers to Selected Exercises Index.

531 citations


Proceedings ArticleDOI
01 Dec 1973
TL;DR: The conditions of applicability of stochastic approximation algorithms that minimize a mean-square error criterion for identification of a linear discrete-time stationary system without dynamical numerator are presented.
Abstract: This study presents the conditions of applicability of stochastic approximation algorithms that minimize a mean-square error criterion for identification of a linear discrete-time stationary system without dynamical numerator. The acceleration of the convergence is discussed. Then a tentative is outlined to overcome the previous requirement of states accessibility.

431 citations


Journal ArticleDOI
TL;DR: The form of the unit threshold likelihood ratio receiver in the detection of a known deterministic signal in additive sirp noise is shown to be a correlation receiver or a matched filter.
Abstract: The n th-order characteristic functions (cf) of spherically-invariant random processes (sirp) with zero means are defined as cf, which are functions of n th-order quadratic forms of arbitrary positive definite matrices p . Every n th-order spherically-invariant characteristic function (sicf) is represented as a weighted Lebesgue-Stieltjes integral transform of an arbitrary univariate probability distribution function F(\cdot) on [0,\infty) . Furthermore, every n th-order sicf has a corresponding spherically-invariant probability density (sipd). Then we show that every n th-order sicf (or sipd) is a random mixture of a n th-order Gaussian cf [or probability density]. The randomization is performed on u^2 \rho , where u is a random variable (tv) specified by the F(\cdot) function. Examples of sirp are given. Relations to previously known results are discussed. Various expectation properties of Gaussian random processes are valid for sirp. Related conditional expectation, mean-square estimation, semMndependence, martingale, and closure properties are given. Finally, the form of the unit threshold likelihood ratio receiver in the detection of a known deterministic signal in additive sirp noise is shown to be a correlation receiver or a matched filter. The associated false-alarm and detection probabilities are expressed in closed forms.

339 citations


Journal ArticleDOI
TL;DR: In this paper, the general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias, characterized by random pauses between jumps, with a common pausing time distributionψ(t).
Abstract: The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between jumps, with a common pausing time distributionψ(t). An analytic solution in the form of an inverse Laplace transform for P(l, t), the probability of a walker being atl at timet if it started atlo att=0, is obtained in the presence of completely absorbing boundaries. Numerical results for P(l, t) are presented for characteristically different ψ(t), including one which leads to a non-Gaussian behavior for P(l, t) even for larget. Asymptotic results are obtained for the number of surviving walkers and the mean 〈l〉 showing the effect of the absorption at the boundary.

313 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that stock price changes are more or less indistinguishable from white noise, or, at the least, their expected percentage movements constitute a driftless random walk (or random walk with mean drift specifiable in terms of an interest factor appropriate to the stock's variability or riskiness).
Abstract: Even the best investors seem to find it hard to do better than the comprehensive common-stock averages, or better on the average than random selection among stocks of comparable variability. Examination of historical samples of percentage changes in a stock's price show that, when these relative price changes are properly adjusted for expected dividends paid out, they are more or less indistinguishable from white noise, or, at the least, their expected percentage movements constitute a driftless random walk (or random walk with mean drift specifiable in terms of an interest factor appropriate to the stock's variability or riskiness). The present contribution shows that such observable patterns can be deduced rigorously from a model which hypothesizes that a stock's present price is set at the expected discounted value of its future dividends, where the future dividends are supposed to be random variables generated according to any general (but known) stochastic process. This fundamental theorem follows by an easy superposition applied to the 1965 Samuelson theorem that properly anticipated futures prices fluctuate randomly -- i.e., constitute a martingale sequence, or a generalized martingale with specifiable mean drift. Examples demonstrate that even when the economy is not free to wander randomly, intelligent speculation is able to whiten the spectrum of observed stock-price changes. A subset of investors might have better information or modes of analysis and get above average gains in the random-walk model; and the model's underlying probabilities could be shaped by fundamentalists' economic forces.

226 citations


Journal ArticleDOI
TL;DR: The theory presented in this paper is based to a large extent on recent results of the author concerning logarithmic concave measures on two stochastic programming decision models, where the solvability of the second stage problem only with a prescribed (high) probability is required.
Abstract: Two stochastic programming decision models are presented. In the first one, we use probabilistic constraints, and constraints involving conditional expectations further incorporate penalties into the objective. The probabilistic constraint prescribes a lower bound for the probability of simultaneous occurrence of events, the number of which can be infinite in which case stochastic processes are involved. The second one is a variant of the model: two-stage programming under uncertainty, where we require the solvability of the second stage problem only with a prescribed (high) probability. The theory presented in this paper is based to a large extent on recent results of the author concerning logarithmic concave measures.

182 citations


Book
01 Jan 1973

110 citations



Journal ArticleDOI
TL;DR: A technique for the digital simulation of multicorrelated Gaussian random processes is described, based upon generating discrete frequency functions which correspond to the Fourier transform of the random processes and then using the fast Fourier Transform algorithm to obtain the actual random processes.
Abstract: A technique for the digital simulation of multicorrelated Gaussian random processes is described. This technique is based upon generating discrete frequency functions which correspond to the Fourier transform of the random processes and then using the fast Fourier transform (FFT) algorithm to obtain the actual random processes. The main advantage of this method over other methods is computation time; it appears to be more than an order of magnitude faster than present methods of simulation. One of the main uses of multicorrelated simulated random processes is in solving nonlinear random vibration problems by numerical integration of the governing differential equations. [This research is supported in part by NASA.]

78 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that an optimal stationary stochastic program can be sustained by a stochastically equilibria, in which at each date the optimal production decisions maximize expected intertemporal profit, and the optimal aggregate consumption vector has minimum cost among all aggregate consumption vectors yielding no less (social) utility.

Journal ArticleDOI
TL;DR: In this paper, the authors examined the autocorrelation function for a particular integrated moving average process and discussed the implications for model identification for nonstationary stochastic processes, where the generating process is non-stationary.
Abstract: SUMMARY Little is known about the behaviour of the sample autocorrelation function when the generating stochastic process is nonstationary. In this note the behaviour of the sample autocorrelation function for a particular integrated moving average process is examined and the implications for model identification are discussed.

Journal ArticleDOI
TL;DR: In this article, it was shown that least-squares filtered and smoothed estimates of a random process given observations of another colored noise process can be expressed as certain linear combinations of the state vector of the so-called innovations representation (IR) of the observed process.
Abstract: We show that least-squares filtered and smoothed estimates of a random process given observations of another colored noise process can be expressed as certain linear combinations of the state vector of the so-called innovations representation (IR) of the observed process. The IR of a process is a representation of it as the response of a causal and causally invertible linear filter to a white-noise "innovations" process. For nonstationary colored noise processes, the IR may not always exist and a major part of this paper is devoted to the problem of identifying a proper class of such processes and of giving effective recursive algorithms for their determination. The IR can be expressed either in terms of the parameters of a known lumped model for the process or in terms of its covariance function. In the first case, our results on estimation encompass most of those found in the previous literature on the subject; in the second case, there seems to be no prior literature. Finally, we may note that our proofs rely on, and exploit in both directions, the intimate relation that exists between least-squares estimation and the innovations representation.

Journal ArticleDOI
TL;DR: In this article, a synopsis of self-similar stochastic processes with special emphasis on fractional Brownian motion and noises is presented, and a step-by-step procedure is included for convenience of application.
Abstract: A synopsis of self-similar stochastic processes is presented with special emphasis on fractional Brownian motion and noises. Detailed analysis and discussion are on the approach of ‘fast fractional Gaussian noises’ by Mandelbrot and his collaborators. Modifications are incorporated and parameter selection criteria suggested so that the method becomes simpler, more flexible to use, and easier to follow. A step-by-step procedure is included for convenience of application.

Journal ArticleDOI
TL;DR: A channel with arbitrarily varying channel probability functions in the presence of a noiseless feedback channel is studied and its capacity is determined by proving a coding theorem and its strong converse and a formula for the zero-error capacity is obtained.
Abstract: In this article we study a channel with arbitrarily varying channel probability functions in the presence of a noiseless feedback channel (a.v.ch.f.). We determine its capacity by proving a coding theorem and its strong converse. Our proof of the coding theorem is constructive; we give explicitly a coding scheme which performs at any rate below the capacity with an arbitrarily small decoding error probability. The proof makes use of a new method ([1]) to prove the coding theorem for discrete memoryless channels with noiseless feedback (d.m.c.f.). It was emphasized in [1] that the method is not based on random coding or maximal coding ideas, and it is this fact that makes it particularly suited for proving coding theorems for certain systems of channels with noiseless feedback. As a consequence of our results we obtain a formula for the zero-error capacity of a d.m.c.f., which was conjectured by Shannon ([8], p. 19).

Journal ArticleDOI
TL;DR: It is shown that (with the conditions imposed) the optimal control is linear in the observed data and can be determined by solving a deterministic problem with a similar dynamic structure.

Journal ArticleDOI
TL;DR: In this article, the reliability and availability characteristics of a 2-unit cold standby system with a single repair facility are analyzed under the assumption that the failure and the repair times are both generally distributed.
Abstract: The reliability and the availability characteristics of a 2-unit cold standby system with a single repair facility are analyzed under the assumption that the failure and the repair times are both generally distributed. System breakdown occurs when the operating unit fails while the other unit is undergoing repair. The system is characterized by the probability of being up or down. Integral equations corresponding to different initial conditions are set up by identifying suitable regenerative stochastic processes. The probability of the first passage to the down-state starting from specified initial conditions is obtained by the same method. An explicit expression for a Laplace Transform of the probability density function (pdf) of the downtime during an arbitrary time interval is obtained when the repair time is exponentially distributed. A general method is suggested for the calculation of the moments of the downtime when the repair time is arbitrarily distributed.

Journal ArticleDOI
01 Jan 1973
TL;DR: A method is suggested for extending the updating schemes known for the P model to the S model, where the environment's output can lie in the interval [0,1], and a class of optimal nonlinear schemes for the Smodel is derived.
Abstract: The performance of variable-structure stochastic automata in stationary random environments has been extensively studied for the case when the environment's response is 0 or 1 (P model). A method is suggested for extending the updating schemes known for the P model to the S model, where the environment's output can lie in the interval [0,1], and a class of optimal nonlinear schemes for the S model is derived. Computer simulations reveal the superior performance of the S model in multimodal search even when the bounds on the performance function are unknown.

Journal ArticleDOI
TL;DR: The theory of constructing observer–estimators for linear, continuous-time systems is described and the case that some observations are noise-free while others are noisy is considered.
Abstract: This paper describes the theory of constructing observer–estimators for linear, continuous-time systems. Both deterministic and stochastic cases are considered; in particular, the case that some observations are noise-free while others are noisy is considered. Asymptotic properties for both time-varying and time-invariant systems are analyzed and the influence of observability and detectability assumptions is considered.

Journal ArticleDOI
TL;DR: In this article, the photospheric motion at one point on the Sun is shown to have the characteristics of a narrow-band random process, with a characteristic correlation time of 23 min and a mean power spectrum that is a smooth, single-peaked function centered at 3.4 mHz.
Abstract: Four Mt. Wilson measurements (T>4 h) of the photospheric motion at one point on the Sun are shown to have the characteristics of a narrow-band random process. The motion is shown to have a characteristic correlation time of 23 min and a mean power spectrum that is a smooth, single-peaked function centered at 3.4 mHz. In order to make this classification we use the analytic signal to estimate the amplitude, phase, and frequency as functions of time. The power spectrum analysis differs from the common approaches in that it uses the theoretical expression for the mean spectrum for a sequence of random pulses. Because of the random nature of the motion, we doubt the existence of more than one eigenfrequency characteristic of the photosphere as a whole. Likewise, any description of the observed motion in terms of simple deterministic functions will be inadequate for the data used here.

Journal ArticleDOI
TL;DR: In this article, a solution procedure for discrete stochastic programs with recourse linear programs under uncertainty is presented, in which the m-dimensional space in which each combination of the discrete values is a lattice point is used to delete infeasible points from the space.
Abstract: This paper presents a solution procedure for discrete stochastic programs with recourse linear programs under uncertainty. It views the m stochastic elements of the requirements vector as an m-dimensional space in which each combination of the discrete values is a lattice point. For a given second-stage basis, certain of the lattice points are feasible. A procedure is presented to delete infeasible points from the space. Thus, the aggregate probability associated with points feasible for this basis can be enumerated, and used to weight the vector of dual variables defined by the basis. Finally, the paper presents a systematic procedure for changing optimal bases so that a feasible and optimal basis is found for every lattice point.

Journal ArticleDOI
TL;DR: In this paper, a variety of analytical approximations applicable to stationary random processes is extended to non-stationary random processes, with the aid of numerical examples, and the merits of each approximation are examined by comparing with the results of simulation.

Journal ArticleDOI
TL;DR: A dynamic programming model with a physical equation and a stochastic recursive equation is developed to find the optimum operational policy of a single multipurpose surface reservoir.
Abstract: The main objective of this paper is to present a stockastic dynamic programming model useful in determining the optimal operating policy of a single multipurpose surface reservoir. It is the unreliability of forecasting the amount of future streamflow which makes the problem of a reservoir operation a stochastic process. In this paper the stochastic nature of the streamflow is taken into account by considering the correlation between the streamflows of each pair of consecutive time intervals. This interdependence is used to calculate the probability of transition from a given state and stage to its succeeding ones. A dynamic programming model with a physical equation and a stochastic recursive equation is developed to find the optimum operational policy. For illustrative purposes, the model is applied to a real surface water reservoir system.

Journal ArticleDOI
TL;DR: The present paper is a generalization of as discussed by the authors, where the authors prove Donsker's theorem for independent, not necessarily identically distributed random variables satisfying a mixing condition, in the space of continuous functions.
Abstract: According to Donsker's theorem XN D> W where W is standard Brownian motion on [0, 1]. (see [1, p. 137]). A similar theorem can be formulated in the space C of continuous functions ([1, p. 68]). Donsker's theorem has been generalized in many directions. Two of them will be taken up in the present paper. The first, due to Prohorov E9], deals with independent, not necessarily identically distributed random variables. Prohorov's theorem says that the properly defined random functions XN converge in distribution to standard Brownian motion if, and only if, the ~, satisfy the Lindeberg condition. (For the details see E2, p. 452], El, p. 77, Problem 1 and p. 143, Problem 7].) The second generalization, due to Billingsley El, p. 177] is concerned with strict sense stationary processes satisfying a mixing condition. In the present paper we shall prove Donsker's theorem for not necessarily identically distributed random variables satisfying a mixing condition. For such random variables the second-named author has proved theorems of a somewhat different character E10].


BookDOI
01 Jan 1973
TL;DR: Some limit theorems for a queueing system with absolute priority in heavy traffic are given in this article, where the authors consider the case of continuous processes with independent increments on a Markov chain.
Abstract: Some limit theorems for a queueing system with absolute priority in heavy traffic.- On certain problems of uniform distribution of real sequences.- Norms of Gaussian sample functions.- On a new approach to Markov processes.- Limit theorems for linear combinations of order statistics.- Some estimates of the rate of convergence in multidimensional limit theorems for homogeneous Markov processes.- Expectation semigroup of a cascade process and a limit theorem.- Potential theory of symmetric markov processes and its applications.- Hilbert space methods in classical problems of mathematical statistics.- On the martingale aproach to statistical problems for stochastic processes with boundary conditions.- Probabilities of the first exit for continuous processes with independent increments on a markov chain.- Noncommutative analogues of the Cramer-Rao inequality in the quantum measurement theory.- Test of hypotheses for distributions with monotone likelihood ratio: case of vector valued parameter.- Criteria of absolute continuity of measures corresponding to multivariate point processes.- Normal numbers and ergodic theory.- On multitype branching processes with immigration.- Statistics of stochastic processes with jumps.- Evolution asymptotique des temps d'arret et des temps de sejour lies aux trajectoires de certaines fonctions aleatoires gaussiennes.- Asymptotic enlarging of semi-markov processes with an arbitrary state space.- The method of accompanying infinitely divisible distributions.- Optimal stopping of controlled diffusion process.- Additive arithmetic functions and Brownian motion.- Asymptotic behavior of the fisher information contained in additive statistics.- Nonlinear functionals of gaussian stationary processes and their applications.- Stationary matrices of probabilities for stochastic supermatrix.- An estimate of the remainder term in the multidimensional central limit theorem.- A remark on the non-linear Dirichlet problem of branching markov processes.- Some remarks on stochastic optimal controls.- On stationary linear processes with Markovian property.- Some limit theorems for the maximum of normalized sums of weakly dependent random variables.- Non-uniform estimate in the central limit theorem in a separable Hilbert space.- Generalized diffusion processes.- Semifields and probability theory.- Convergence to diffusion processes for a class of Markov chains related to population genetics.- Random operators in a Hilbert space.- Bernoulli shifts on groups and decreasing sequences of partitions.- On the second order asymptotic efficiencies of estimators.- On the relaxed solutions of a certain stochastic differential equation.- On limit theorems for non-critical Galton-Watson processes with EZ1logZ1=?.- Construction of diffusion processes by means of poisson Point process of Brownian excursions.- Non-anticipating solutions of stochastic equations.- A stochastic maximum principle in control problems with discrete time.- Selection of variables in multiple regression analysis.

Journal ArticleDOI
TL;DR: In this article, series representations are obtained for the entire class of measurable, second order stochastic processes defined on any interval of the real line, including all earlier representations; they suggest a notion of smoothness that generalizes well-known continuity notions.

Journal ArticleDOI
TL;DR: In this article, the vibration of a two-degree-of-freedom elastic system due to wind loading is investigated by a Monte Carlo technique, and the response analysis is performed in time domain by numerically simulating the resulting wind forces.
Abstract: The vibration of a two-degree-of-freedom elastic system due to wind loading is investigated by a Monte Carlo technique. The response analysis is performed in time domain by numerically simulating the resulting wind forces. The fluctuating wind velocity field is idealized as a stationary Gaussian random process with mean zero. For wind loading and response analysis, both across-wind and along-wind directions are considered. The results are used to study the effect of mechanical and aerodynamic parameters of the systems and to compare the current formulation with the approximate treatment commonly used.

Journal ArticleDOI
C. E. Newman1
TL;DR: In this article, a generalization of this formalism is developed from two different points of view, a Fokker-Planck approach and a quasilinear approach, leading to the same equation for the evolution of the particle distribution.
Abstract: The problem of a system of charged particles moving in a random force field and its application to the study of turbulent plasmas are discussed. We point out the inapplicability of existing formalisms to many cases of astrophysical interest. A generalization of this formalism is developed from two different points of view—a Fokker‐Planck approach and a quasilinear approach. Both approaches lead to the same equation for the evolution of the particle distribution; this equation has the form of a Fokker‐Planck equation, but the terms describing the effects of the random field do not retain the interpretation which they have in usual Fokker‐Planck development. It is shown that this equation reduces to a quasilinear diffusion equation when the random field is electromagnetic.