Discrete Random Process Research Articles

Exchange-Rate Variability, Real and Monetary Shocks, and the Degree of Capital Mobility Under Rational Expectations

Much of the recent literature on flexible exchange rates has shared two characteristics: first, it has emphasized the critical role of expectations in exchange-rate determination, and second, it has, either explicitly or implicitly, been concerned with exchange-rate volatility or variability.' In light of these common threads, it is somewhat surprising that, with the exception of Barro [1978], none of these models has explicitly incorporated uncertainty. Without this, the forecasting problem, which is at the heart of expectations formation, cannot be adequately analyzed, and it is impossible to describe rigorously what is meant by exchange-rate variability. Consequently, comparative-statics results relating changes in structural parameters to changes in exchange-rate variability cannot be rigorously derived. This paper develops a model of a small, open economy under flexible exchange rates that explicitly incorporates uncertainty. The forecasting problem at the heart of expectational formation can thus be explicitly modeled by assuming that agents make unbiased forecasts based upon knowledge that includes the structure of the model; that is, agents are assumed to have rational expectations in the sense of Muth [1961]. Also, the explicit discrete random process that describes exchange-rate behavior over time is derived, and comparative-statics results about the characteristics of this process are developed. The most interesting result obtained is that increases in the degree of capital mobility, i.e., increases in the degree of portfolio sensitivity to changes in anticipated relative asset yields, decrease that portion of exchange-rate variability attributable to real shocks, but increase that portion attributable to monetary shocks. Another result is that correlations between the trade balance, relative price changes, and spot and forward rates depend on the relative importance of real and monetary shocks. In a sense, this paper extends the pioneering work of Barro [1978], which also explicitly incorporated uncertainty. Our model, though, focuses on three assumptions not incorporated in Barro's model but

On the strong information singularity of certain stationary processes (Corresp.)

In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation. In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation. In an exploratory paper, T. Berger studied discrete random In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation.

Discrete properties of random surfaces

The problem of measuring surface parameters of tribological importance such as peak curvature has been considerably simplified. Using discrete random process analysis tribological parameters of a wide range of surfaces can now be expressed and measured in terms of just two points on the measured autocorrelation function and the r.m.s. value of the surface. In addition, the contribution each ' scale of size ’ of asperity makes to an overall parameter can be assessed quantitatively. Many new expressions relating tribological parameters to the autocorrelation function have been derived using a limiting procedure which produces results entirely consistent with equivalent continuous theory. Using this theory it is now possible to predict ‘gap’ parameters between two surfaces in contact in terms of simple additive parameters of each surface. Finally a new statistical model of the surface has been developed which encompasses many types of engineering surface.

THE PHASE PROPERTY OF A FINITE REALIZATION OF RANDOM NOISE AND ITS INFLUENCE ON DECONVOLUTION *

AbstractA finite realization of a discrete random noise process may be considered as a one‐sided energy signal. Its phase property can then be described by means of the center position. The samples of such a realization are the components of a random signal vector and the center position is therefore a random variable. A statistical analysis shows that the expected value of the center position equals half the time duration of the realization. This implies that the Z‐transform of the realization may be expected to have an equal number of poles and zeros inside and outside the unit circle. The standard deviation from the expected value of the center position is shown to depend on the time duration of the realization and on the autocorrelation of the process. It follows that, for processes that can be described by the convolution of a white series and a disturbance wavelet, the center position is independent of the phase property of the wavelet. A conclusion based on these results is that the homomorphic technique of wavelet estimation through cepstrum stacking must give questionable outcomes. Another conclusion is that the super‐position of a realization of random noise on a minimum phase wavelet will in general give a mixed phase resulting signal. It is pointed out that schemes for the derivation of deconvolution filters do not take account of this phenomenon.

A test for whiteness

In this note a test to check if a discrete random process is white or not is proposed. Its reliability is compared to that of another test commonly used in system identification [1], filtering [3], etc.

The maximum mutual information between two random processes

Given two discrete random processes, what is the largest possible average mutual information between them? An application of the Ornstein—Sinai theorem of ergodic theory is used to show that if the processes are ergodic, then there exists a pair process with the given processes as coordinates such that the average mutual information between the coordinates is the maximum possible value, the smaller of the two entropy rates. As an application and interpretation, the information—theoretic analog of a recently developed distance measure on random processes is defined and the topologies induced by the two distance measures on spaces of random processes are compared and contrasted. This demonstrates a fundamental relation between mutual information and mutual approximation of random processes.

The ergodic decomposition of stationary discrete random processes

The ergodic decomposition is discussed, and a version focusing on the structure of individual sample functions of stationary processes is proved for the special case of discrete-time random processes with discrete alphabets. The result is stronger in this case than the usual theorem, and the proof is both intuitive and simple. Estimation-theoretic and information-theoretic interpretations are developed and applied to prove existence theorems for universal source codes, both noiseless and with a fidelity criterion.

A Geometric Theory of Intersymbol Interference

A linear-space geometric theory of intersymbol interference is introduced in this paper. An equivalence between the structure of intersymbol interference and a wide-sense stationary discrete random process is demonstrated and exploited to demonstrate the equivalence of zero-forcing (decision-feedback) equalization to minimum mean-square error linear interpolation (prediction) of a random process. This equivalence is used to quickly derive the properties of these equalizers and give them additional geometric interpretation. Results from prediction theory are used to develop practical computational methods of determining the tap-gains of the infinite equalizers for both rational and nonrational channel power spectra. Finally, the theory of reproducing kernel Hilbert spaces is used to develop a theory of equalization for nonstationary channels with nonstationary noise.

Computational algorithms for discrete detection and likelihood ratio computation

A pseudo Bayes approach to likelihood ratio determination for discrete time random processes imbedded in Gaussian noise is presented. The resulting computationally feasible algorithm for the likelihood ratio is expressed as a function of the one-step prediction conditional mean estimate of the message, suggesting an estimator-correlator type structure for the optimum digital detector. The use of smoothed or iterated conditional mean estimates in the pseudo Bayes detection scheme is also considered. The likelihood ratio formulas for continuous random signals in Gaussian noise are derived from the discrete results as a limiting case. Examples indicate the efficacy of the method.

On a Method for Estimating Power Spectra

In many engineering problems, the power spectra of random signals have to be measured. In this paper, a method for estimating the unknown parameters in the power spectral density functions of discrete random processes is presented. The spectral density functions are represented by a ratio of two real polynomials in z. It is assumed that only a finite record of the random process is observed and no knowledge about its probability distribution is required. An estimation procedure is developed. It is shown that the denominator parameter estimates can be first obtained. Then the numerator parameters are estimated by filtering the poles from the spectrum. A four parameter example is given to demonstrate the method.

A New Approach to Green's Function of a Particle in One Dimension

The Schrodinger equation of a particle moving in one dimension in a given static potential is solved. The density matrix corresponds to a Laplace transform of Green's function and satisfies a diffusion-type equation whlch may be regarded as representing a certain limit of a discrete Markoffian random walk process. The space occupied by the system is divided into nonoverlapping regions; in each, the Sehrodinger equation can be solved. A path concept is introduced to synthesrze the solutions for the regions and formulate the one- particle Green's function. The energy bands of a one-dimensional infinite system are discussed. The condition for the existence of the impurity band is derived, which qualitatively supports Anderson's theory on the absence of diffusion in certain random lattices.

Discrete random Processes

If a variable assumes the discrete valuesxj(j= 1, 2, 3, …) with specified probabilitiesf(xj), wheref(xj) it is said to be adiscrete random variable. If a discrete random variable is also a function of a continuous (non-random) variable, for convenience usually assumed to be ‘time’, it is called adiscrete random process.A class of discrete random processes of particular interest to the pure mathematician and to the mathematical statistician has been calledstochastically definiteby Kolmogoroff (1931). Such a random process is distinguished by the fact that the probability that the random variable concerned assumes a given valuenat timetdepends only on the valuemassumed by the variable at times(s < t) and not on the values assumed at any intermediate or earlier points of time. This circumstance is allowed for by writing the probability of the valuenat timetin the form Pmn(s, t).

The probability of decrements from a population. A study in discrete random processes

Abstract The statistician, whether he be biologist, actuary, or worker III the social sciences, often requires to measure the intensity of some decremental force, or forces, acting on an observed body of individuals. The latter may be human beings, animals, or even inanimate objects (e. g. aircraft), and the decrements considered may range from marriage to‘write-off’. N evertheless the assumptions made are, for the most part, analogous and the technique adopted is broadly the same in every case: the decrements are assumed to be random (in the technical sense) and they are related to the body in which they have occurred (the so-called‘exposed to risk’) in order to obtain a relative frequency, or‘rate’, of decrement.