Showing papers on "Stochastic process published in 1975"

Dynamics Of Structures

Abstract: Part 1 Single-degree-of-freedom systems: overview of structural dynamics analysis of free vibrations response to harmonic loading response to periodic loading response to impulse loading responses to general dynamic loading - step by step methods, superposition methods generalized single degree-of-freedom systems. Part 2 Multi-degree-of-freedom systems: formulation of the MDOF equations of motion evaluation of structural-property matrices undamped free vibrations analysis of dynamic response using superposition vibration analysis by matrix iteration selection of dynamic degrees of freedom analysis of MDOF dynamic response - step by step methods variational formulation of the equations of motion. Part 3 Distributed parameter systems: partial differential equations of motion analysis of undamped free vibrations analysis if dynamic response. Part 4 Random vibrations: probability theory random processes stochastic response of linear SDOF systems stochastic response of non-linear MDOF systems. Part 5 Earthquake engineering: seismological background free-field surface ground motions deterministic structural response - including soil-structure interaction stochastic structural response.

A survey of design methods for failure detection in dynamic systems

Abstract: A number of methods for detecting abrupt changes (such as failures) in stochastic dynamical systems are surveyed. The class of linear systems is concentrated on but the basic concepts, if not the detailed analyses, carry over to other classes of systems. The methods surveyed range from the design of specific failure-sensitive filters, to the use of statistical tests on filter innovations, to the development of jump process formulations. Tradeoffs in complexity versus performance are discussed.

Markovian Representation of Stochastic Processes by Canonical Variables

Abstract: The structure of the information interface between the future and the past of a discrete-time stochastic process is analyzed by using the concepts of canonical correlation analysis. Two extreme Markovian representations are obtained with states defined by the sets of canonical variables which represent the past information projected on the future and the future information projected on the past, respectively. The result completely clarifies the probabilistic structure of the Faurre algorithm of realization of stochastic systems. By an extension of the basic result the Ho–Kalman algorithm of realization of general systems is also given a stochastic interpretation.

A Strong Law of Large Numbers for Random Compact Sets

Abstract: : In the study of probabilities on geometrical objects, there have been some recent attempts to formulate general theories of random sets, notably by Kendall and Matheron. It is the purpose here to make a contribution in this direction by demonstrating the existence of a strong law of large numbers for random sets taking values in the class of compact subsets of (R sup n). The result is proved first under the assumption of convexity and then extended to the general case.

Simulating Stable Stochastic Systems: III. Regenerative Processes and Discrete-Event Simulations

Abstract: This paper shows that a previously developed technique for analyzing simulations of GI/G/s queues and Markov chains applies to discrete-event simulations that can be modeled as regenerative processes. It is possible to address questions of simulation run duration and of starting and stopping simulations because of the existence of a random grouping of observations that produces independent identically distributed blocks in the course of the simulation. This grouping allows one to obtain confidence intervals for a general function of the steady-state distribution of the process being simulated and for the asymptotic cost per unit time. The technique is illustrated with a simulation of a retail inventory distribution system.

Time-reversibility of linear stochastic processes

Abstract: Time-reversibility is defined for a process X(t) as the property that {X(t), - - -, X(t.)} and {X(- t), - -, X(- t.)} have the same joint probability distribution. It is shown that, for discrete mixed autoregressive moving-average processes, this is a unique property of Gaussian processes. TIME-REVERSIBILITY; SHOT NOISE; CHARACTERISATIONS OF THE NORMAL DISTRIBUTION; TIME SERIES; STOCHASTIC PROCESSES

On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars

Abstract: This paper studies several geometric aspects of the Poisson and Gaussian random fields approximating Burgers k−2 and Kolmogorov .

States of classical statistical mechanical systems of infinitely many particles. I

Abstract: We study a general mathematical model of a classical system of infinitely many point particles. The space X of infinite particle configurations is equipped with a natural topology as well as a measurable structure related to it. It is also connected with a family {XA} of local spaces of finite configurations indexed by bounded open sets A in the one-particle space E. A theorem analogous to Kolmogoroff's fundamental theorem for stochastic processes is proved, according to which a consistent family {μA} of local probability measures μAdefined on the XAgives rise to a unique probability measure μ on X. We also study the problem of integral representation for positive linear forms defined over some linear space of real functions on X. We prove that a positive linear form F(f), defined for functions f in the class C+P, admits a uniquely determined integral representation F(f)=∝ f (ξ) dμ, where μ is a probability measure over X.

Feedback between stationary stochastic processes

Abstract: A simple formulation is given for the notion of feedback between two stationary stochastic processes in terms of the canonical representation of the joint process. The definition presented here has an equivalent formulation in terms of filtering theory, and provides statistical criteria for the detection of feedback. A simulation example is presented and an application to the United Kingdom unemployment-gross domestic product relation is described.

A Generalization of Ornstein's $\bar d$ Distance with Applications to Information Theory

Abstract: Ornstein's $\bar{d}$ distance between finite alphabet discrete-time random processes is generalized in a natural way to discrete-time random processes having separable metric spaces for alphabets. As an application, several new results are obtained on the information theoretic problem of source coding with a fidelity criterion (information transmission at rates below capacity) when the source statistics are inaccurately or incompletely known. Two examples of evaluation and bounding of the process distance are presented: (i) the $\bar{d}$ distance between two binary Bernoulli shifts, and (ii) the process distance between two stationary Gaussian time series with an alphabet metric $|x - y|$.

Field-theoretic techniques and critical dynamics. I. Ginzburg-Landau stochastic models without energy conservation

Abstract: The critical dynamics of a stochastic Ginzburg-Landau model of an $N$-component order parameter coupled to a conserved-energy-density field is studied with the help of field-theoretical techniques introduced in previous work. Our results essentially confirm and refine upon those of Halperin, Hohenberg, and Ma. Scaling laws are derived (whenever they hold). A better knowledge of the domain structure of the ($N, d$) plane and the corresponding critical exponents is obtained, in particular one additional region is shown to be present. Stability criteria lead to a characterization of the leading corrections to dynamical scaling by extra exponents which, except for one of them, are related to known static exponents.

Weak Convergence of Empirical Distribution Functions of Random Variables Subject to Perturbations and Scale Factors.

Abstract: : The weak convergence of empirical distribution functions of independent identically distributed random variables is well-known and has been studied by several authors notably Doob (1949) and Donsker (1951. In an earlier paper, Sethuraman and Rao (1970), while studying the asympototic efficiencies of tests based on spacings the authors found a need to study the weak convergence of empirical distribution functions of random variables subject to random perturbations and scale factors. The authors study this problem in this paper. The material of this paper is divided into two sections. Section 2 treats the aforementioned problem and is independent of Sethuraman and Rao (1970). Section 3 shows how one can use the weak convergence results of Section 1 in problems connected with spacings thus relating the present work to Sethuraman and Rao (1970). (Author)

Some Distributions Associated with Bose-Einstein Statistics

Abstract: This paper examines a stochastic process for Bose-Einstein statistics that is based on Gibrat's Law (roughly: the probability of a new occurrence of an event is proportional to the number of times it has occurred previously). From the necessary conditions for the steady state of the process are derived, under two slightly different sets of boundary conditions, the geometric distribution and the Yule distribution, respectively. The latter derivation provides a simpler method than the one earlier proposed by Hill [J. Amer. Statist. Ass. (1974) 69, 1017-1026] for obtaining the Pareto Law (a limiting case of the Yule distribution) from Bose-Einstein statistics. The stochastic process is applied to the phenomena of city sizes and growth.

A Stochastic Analysis of Extreme Droughts

Abstract: A stochastic analysis is performed on the extreme drought duration defined to be the maximum dry interval for a point rainfall process. The assumptions underlying previous analyses are generalized to those of a nonhomogeneous Poisson process. Analytical results, which seem intractable in general, are derived for two particular forms of the intensity function of the Poisson process; for a general intensity function, simulation is recommended. Next, small time intervals such as the growing season of a crop are considered so that the assumption of a homogeneous Poisson rainfall process may be made and the effect of parameter estimation on the theoretical results may be studied qualitatively. To illustrate this point, four estimates of the intensity parameter are calculated by using precipitation data from Chicago and Austin. A good agreement is found between the theoretical and empirical distribution functions for the two parameter estimates calculated by use of the model developed in this investigation; on the other hand, a substantial bias is present for parameters calculated directly from the data. Finally, an approach is schematically indicated to extend the model to regional droughts by using stochastic superposition.

A theory of spontaneous fluctuations in macroscopic systems

Abstract: A theory is developed for the dynamics of spontaneous fluctuations in systems which, on the average, obey nonlinear transport equations. The theory is a generalization of the Ornstein–Uhlenbeck theory of near equilibrium fluctuations (Langevin‐type theories) and yields a stochastic process which is nonstationary with a Gaussian conditional probability. The three assumptions on which the theory is based are predominately kinetic in nature and in order to apply the theory it is necessary to formulate rate equations in terms of elementary events. Examples of this are given for nonlinear chemical reactions and the complete nonlinear Boltzmann equation.

Process definitions of distortion-rate functions and source coding theorems

Abstract: The standard definition of the distortion-rate function involves a limit of information-tbeoretic minimizations over distributions of random vectors. Several alternative definitions, each involving a single minimization over random processes, are presented here and verified. These definitions parallel Khinchine's process definition of channel capacity, provide a new interpretation of block and nonblock source coding (with a fidelity criterion) theorems in terms of optimal stochastic codes, and provide a comparison between the optimal performance theoretically attainable (OPTA) using block and nonblock source codes. Coupling the process definitions with recently developed bounding techniques provides a new and simple proof of the block source coding theorem for ergodic sources.

Onsager-Machlup Function for one dimensional nonlinear diffusion processes

Abstract: Using Ito stochastic differential equations to describe stochastic processes the Onsager-Machlup Function of a nonlinear diffusion process is calculated. It is shown that for two examples the Onsager-Machlup Function calculated directly as limit of finite dimensional probability densities agrees with the formula derived by using the Ito calculus but differs from a formula given by Graham who used the concept of Langevin equations.

The Comparison Method for Stochastic Processes

Abstract: A relationship between the path structure of two real discrete time stochastic processes is deduced from inequalities between their transition functions. The approach is to define processes equivalent to the two on a common space so that pointwise inequalities are possible. An iterated logarithm type law for random walks is given as a particular application of the general method.

Random temporal variation in selection intensities: One-locus two-allele model

Abstract: The paper formulates a one locus two allele diploid model influenced by temporal random selection intensities. The concept of stochastic local stability for equilibria states is formulated. Necessary and sufficient conditions are derived for the stochastic local stability of the fixation states for non-dominant and dominant traits. A general model, where a fixed polymorphic equilibrium is maintained is investigated. The complete local evolutionary picture is derived and some cases of global convergence are established.

A generalized theory of multiplicative stochastic processes using cumulant techniques

Abstract: The rules for the construction of the nth order cumulant for time−dependent, stochastic, matrices or operators which do not commute with themselves at unequal times are derived. The results are identical with van Kampen’s rules. In the Gaussian case, Kubo’s concept of a generalized Gaussian process is criticized. Under certain conditions Kubo’s idea becomes asymptotically valid, while the same conditions justify use of the author’s earlier delta function theory. A generalized density matrix equation is presented and its behavior during the approach to equilibrium is discussed. A finite correlation time, τc, does not necessarily invalidate a monotonic approach to equilibrium.

Estimation for rotational processes with one degree of freedom--Part I: Introduction and continuous-time processes

Abstract: A class of bilinear estimation problems involving single-degree-of-freedom rotation is formulated and resolved. Continuous-time problems are considered here, and discrete-time analogs will be studied in a second paper. Error criteria, probability densities, and optimal estimates on the circle are studied. An effective synthesis procedure for continuous-time estimation is provided, and a generalization to estimation on arbitrary Abelian Lie groups is included. Applications of these results to a number of practical problems including frequency demodulation will be considered in a third paper.

Stochastic theory for classical and quantum mechanical systems

Abstract: We formulate from first principles a theory of stochastic processes in configuration space. The fundamental equations of the theory are an equation of motion which generalizes Newton's second law and an equation which expresses the condition of conservation of matter. Two types of stochastic motion are possible, both described by the same general equations, but leading in one case to classical Brownian motion behavior and in the other to quantum mechanical behavior. The Schrodinger equation, which is derived here with no further assumption, is thus shown to describe a specific stochastic process. It is explicitly shown that only in the quantum mechanical process does the superposition of probability amplitudes give rise to interference phenomena; moreover, the presence of dissipative forces in the Brownian motion equations invalidates the superposition principle. At no point are any special assumptions made concerning the physical nature of the underlying stochastic medium, although some suggestions are discussed in the last section.

A Minimum Principle for Controlled Jump Processes

Abstract: In Queueing Theory and many other fields problems of control arise for stochastic processes with piecewise constant paths. In this paper the validity of optimality conditions analagous to the Pontryagin Maximum Principle for deterministic control problems is investigated for this type of stochastic process. A minimum principle which involves the conditional jump rate, the conditional state jump distribution, system performance rate, and the conditional expectation of the remaining performance is obtained. The conditional expectation of the remaining performance plays the role of the adjoint variables. This conditional expectation satisfies a type of integral equation and an infinite system of ordinary differential equations.

1 – Stochastic Processes

Abstract: Publisher Summary This chapter presents the Kolmogorov construction of a stochastic process. A stochastic process is a family of random variables defined on a probability space. Given a consistent set of distribution functions, there exists a stochastic process defined on a probability space, such that the given distribution functions are the pint distribution functions of the process. A ν-dimensional countable stochastic process is a sequence of ν-dimensional random variables. The chapter presents an overview of the separable and continuous processes. Any ν-dimensional process is stochastically equivalent to a separable stochastic process. The chapter discusses martingales and stopping times.