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The reflection coefficients at normal incidence are found for a large class of one-dimensionally inhomogeneous or stratified half-spaces, which contain a homogeneous half-space. The formulation of the problem involves a combination of the classical boundary value technique, and the nonclassical principle of invariant imbedding. Solutions are in closed form and expressible in terms of Bessel functions. All results are given in terms of the ratio of the distance between free space and the homogeneous half-space to the wavelength in vacuo. One special case is that of an arbitrary number of layers lying on a homogeneous half-space where the dielectric constant of each layer has a constant gradient. A number of other special cases, limiting cases, and generalizations are developed including one in which the thickness of the top layer obeys a probability distribution.

The modified Hankel transform arises naturally in connection with certain semigroup operations on measures in probability theory. We give a tauberian theorem for this transform when certain higher moments exist. The probabilistic significance of our result is that it translates a regularity condition on the transform into a direct condition on the measure. This complements earlier results by Pitman and Bingham for the trigonometric and the modified Hankel transform respectively.

The asymptotic normal distribution of a sum of dependent random functions of m variables defined on the positive part of the integral lattice is established by the method of moments.

An algorithm is developed for solving a special structured linear-fractional program. The structure under study hasM+L constraints equations,L of which have the property that each variable has at most one nonzero coefficient. The proposed method is similar toDantzig andVan Slyke and, from the basis, a working basis of orderM is derived and is used for pivoting, pricing and inversion which for largeL can be significantly lower order than that of the original system.

The concept of relative information functions by means of a functional equation under suitable boundary conditions is introduced and discussed. Further the relative information is defined in terms of these functions and this idea is extended to the new concept of relative information functions of type (α, β) (α, β>0, α≠β) and relative information of type (α,β) which are the generalizations of the earlier ones. Finally the different types of relative informations of type (α, β) are defined for bivariate distributions and the relations between them are established.

The shadowing function, defined as the probability that a point on a rough surface will be illuminated by radiation from a source at fixed height above the surface, is considered. An exact expression for this probability is derived and evaluated approximately when the surface elevation is a normal random process. The effect of surface correlation is considered and shown to be appreciable. Several numerical computations of the shadowing function are presented.

The main purpose of this paper is to study certain representations of sums of iid k-vector rv's as embeddings in k-dimensional Brownian motion by vectors of stopping times, in extension of Skorohod's scheme [20], and consequent error estimates for weak and strong invariance principles. In particular, letting k→∞ we embed the sample df in the Gaussian process with 2-dimensional time to which it has long been known to converge weakly. We discuss previous sample df embeddings, which have yielded related results; while some of our estimates are slight improvements, the emphasis here will be on the naturality of the embedding per se (although it will be indicated why it is probably far from the final word on the subject.

A METHOD IS PRESENTED TO FIND UPPER AND LOWER BOUNDS OF THE PROBABILITY THAT A SIMPLE STRUCTURE WILL FAIL UNDER A RANDOM LOADING OF EARTHQUAKE TYPE. THE STRUCTURE CONSIDERED IS A ONE STORY BUILDING TREATED AS A SYSTEM OF MASS, SPRING AND DASHPOT WITH SINGLE DEGREE OF FREEDOM. THE FAILURE IS ASSUMED TO OCCUR WHEN THE ABSOLUTE VALUE OF THE DISPLACEMENT OF THE MASS EXCEEDS A CRITICAL VALUE. THE PRESENT METHOD LEADS TO EVALUATION OF LENGTHY INTEGRALS WHICH, HOWEVER, CAN BE PERFORMED NUMERICALLY WITH AN ELECTRONIC DIGITAL COMPUTER AND PRODUCES THE UPPER AND LOWER BOUND CLOSE ENOUGH AT LEAST FOR THE ORDER ESTIMATION OF THE PROBABILITY. /AUTHOR/

One method of probability density estimation is representing the unknown density by an expansion of the form $f(x) = \sum {a_j \phi _j (x)} $ and estimating the coefficients $a_j $ from a sequence of independent observations $\{ X_i \}$. In this paper, we consider estimating Gaussian-type densities, by which we mean that $f(x)$ is the convolution of a Gaussian probability density and an arbitrary distribution. It is shown that the above expansion can always be obtained, with the coefficients $a_j $ expressed as a product of two terms : one a function only of the unknown distribution and the other term dependent on the Gaussian density. Rates of convergence and a comparison with other methods of density estimation are discussed.

Let C n =C n (T) be the Fourier coefficients of a continuous weakly stationary process over (−1/2T,1/2T). We make some corrections of the results on the magnitude of $$\sum\limits_{\left| n \right| > N} {E\left| {C_n } \right|^2 } {\text{ }}(N \to \infty ),$$ , in the author's previous work. We discuss in this connection about the periodicity and non-periodicity properties of weakly stationary processes. We also discuss the approximate Fourier series which approximates the given weakly stationary process and using some results on it we give a sufficient condition for the sample continuity of a weakly stationary process which is an improvement of a result of Cramer and Leadbetter.

In Part 1 the operator calculus of ordered exponentials is discussed, which are called the formal solutions of a partial integro-differential equation∂ψ(x, s)/∂s=A(s)ψ(x, s) and have the properties of semi-groups. In Part 2. we show explicit representation of the ordered exponential and the method how to set up the path integral which is proposed by Feynman as a new formulation of quatum mechanics. This method is applicable for solving a partial integro-differential equation of the above mentioned type and connects the above equation with the probability theory. When the multiple-integration of the path integral is too cumbersome to carry out rigorously, one may adopt the one path approximation, the principle of which is to take a most probable path instead of the summation over all possible paths. We shall state a new approximation of choosing one path in phase space. The method mentioned in this paper is applicable to the problems of non-stationary Markoff processes. Applications to the probability theory are discussed in Part 3 taking various examples. This paper is the revision of the previous paper* and contains new results.

A stoehastic discrete Volterra equation in the form is studied, where ω∈Ω, the supporting set of the probability measure space (Ω, A, P). A theorem is presented which guarantees the existence of a unique random solution of the equation. A random solution is defined to be a discrete parameter second-order stochastic process xn (ω) which satisfies the equation with probability one. An application of the results to a non-linear stochastic control system is also given.

A functional integral representation for the space‐time Hopf characteristic functional is derived from the probability theory for a statistical ensemble of velocity fields that satisfy the Navier‐Stokes equation for boundary‐free incompressible fluid flow. The functional integral representation involves a pair of real vector field integration variables denoted by u and v, and the evaluation of the integral is performed in two steps. First, the integration over the field variable u is effected exactly in the general case by applying methods of explicit functional integration. Second, the resulting functional integral over the field variable v is reduced to a form amenable to specialized analysis by applying a suitable transformation of the integration field variable v → z. Specializing to mathematically defined ``C‐dominant turbulence,'' the final functional integration over the field variable z is effected exactly and yields a characteristic functional of Gaussian form. The two‐point velocity correlation ...

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: М. П. Сапаговас. Решение квазилинейных эллиптических уравнений методом конечных разностей M. P. Sapagovas. Kvazitiesinių eliptinio tipo diferencialinių lygčių sprendimas baigtinių skirtumų metodu

Variational sums for arbitrary processes with stationary, independent increments are shown to converge in probability to certain other such processes as the mesh of a sequence of partitions tend to zero. Under less restrictive conditions, centered variational sums are shown to converge. In cases of divergence, rates of divergence are found. (Author)

A measurable p-function is necessarily either a constant multiple of a standard p-function, or is almost everywhere zero.

Efficient and practical methods of simulating stationary and nonstationary random envelope processes are presented. The stationary envelope processes are simulated by using the fast Fourier transform while the nonstationary envelope processes are simulated as the square root of the sum of a series of cosine functions and a series of sine functions with random phase angles. Typical applications of the envelope simulation are the simulations of peaks and troughs which play an important role in the analyses of the first excursion probability, fatigue and crack propagation. In particular, applications to the crack propagation under random loadings are demonstrated in detail.

The purpose of this note is to give a consequence of a theorem of Rellich [3, p. 6841. Rellich showed that if a sequence of self-adjoint operators H, over a Hilbert space H converges strongly to the self-adjoint operator H in the generalized sense, then the resolutions of the identity E,(X) of H, converge strongly to the resolution of the identity E(h) of H for all h which are not point eigenvalues of H. We will show that the convergence of i?,(h) x to E(h) x is in fact uniform in h for all vectors x belonging to the continuous subspace of the operator H. This result is analogous to a theorem in probability theory, sometimes known as Polya’s theorem, which states that if a sequence of distribution functions F,(x) converges weakly to a continuous distribution function F(x) then the convergence is uniform [1 , p. 2681.

Starting fromLe Cam [1956], it was shown inMichel andPfanzagl [1970] that — under certain regularity conditions — a dominated family of probability measures withEuclidean parameter space behaves approximately like a family of normal distributions, if each probability measure is the independent product of a great number of identical components. It is the purpose of this paper to estimate the accuracy of such a normal approximation.

A new quadrature formula for an important class of Wiener integrals is presented, in which the Wiener integrals are approximated by «-fold integrals with an error Oin~2). The resulting «-fold integrals can then be approximated by ordinary finite sums of remarkably simple structure. An example is given. Introduction. Wiener integrals in function space play a major role in a number of applications in physics and in probability theory, see e.g. (1), (6), (7), (9). A number of remarkable results have been obtained concerning the approximation of these integrals by finite-dimensional integrals (see in particular Cameron (2), as well as (8), (10), and (14)). The resulting n-fold integrals are, in general, difficult to evaluate with any accuracy, and as a consequence the approximation formulas are not of significant practical use. The aim of this paper is to present a new approximation for Wiener integrals accurate enough and simple enough to be of practical interest. Some of the elegant generality of Cameron's work may be lost, but the method is applicable to many functionals which appear in physics, and will furthermore afford an intuitive grasp of the relation between ordinary quadrature and quadrature in a function space. The two main ideas in the approximation method are the following: the Wiener paths are carefully interpolated by a certain family of parabolas, in such a way that all the moments are exactly reproduced; and nonlinear functionals are expanded in a certain Taylor series, with the quadrature formula adjusted so that the first two groups of terms are well approximated. Outline of Goal and Method. Let C be the space of continuous real functions x(i) defined on 0 ^ t ^ 1, with x(0) = 0, and endowed with the Wiener measure W. Let F(x) be a functional on C; our aim is to evaluate

The equation of the vibrating string forced by white noise is formally solved, using stochastic integrals with respect to a plane Brownian motion, and it is proved that a certain process associated to the energy is a martingale. Then Doob's martingale inequality is used to furnish some probability bounds for the energy.

A random field over l p is a stochastic process X(t), where t is an element of l p .It is said to have homogeneous and isotropic increments if E(X(t) − X(s)) 2 is a function of ∥t-s∥. The subject of this work is the spectral theory of such processes. The main results are: a representation of the field as a series of filtered, orthogonal processes with a real time parameter; a representation as a white noise integral over l p ;limit theorems for the average of X over a sphere; and, finally, filtering of the orthogonal components. In particular, we mention: (1) The averages over spheres of increasing dimension converge in quadratic mean for p=2 but not for 0

The connection with a semigroup of nonlinear operators is exploited to prove an existence theorem. A rigorous description of the process is provided which does not depend on any scheme for indexing particles.

The work of a previous paper which examined the propagation of a fatigue crack under random, exponentially distributed, cyclic loadings is extended. The exact recurrence relations for the risk function derived earlier are solved by an improved approximation technique which produces as zero order solution a survival function which is without significant error in most circumstances of practical value. The first corrections to this solution are obtained. It is then pointed out that this solution could be obtained from a very crude approach using mean crack lengths. This latter method can be readily extended to other crack propagation laws and other stress distributions, as is demonstrated in some cases of practical interest. Some remarks are also added on the relationship between crack propagation and the damage accumulation concept.

This paper is concerned with the nonlinear behavior of the second-order phase-locked loop (PLL) in the presence of noise. The loop filter is of the proportional-plus-integral control type. This filter corresponds to the one generally employed for carrier tracking purposes in the implementation of phase-coherent communication systems. The paper is composed essentially of two parts: the first part presents analytical results which pertain to the probability distribution of the phase-error. Since these analytical results are approximations, valid only for certain regions of signal-to-noise ratio, they are complemented by experimental results obtained from simulation of the PLL system in the laboratory. The experimental techniques used to measure the statistical properties of the loop behavior and the corresponding results comprise the second part of the paper. Approximate analytical expressions for the distribution of the system phase-error are first obtained by using the Fokker-Planck apparatus and, secondly, by assuming that the PLL behaves as a very narrow band-pass filter. The range of signal-to-noise ratios for which these approximations are valid is obtained by graphically comparing the analytical expressions to experimentally derived phase-error distributions. In addition, measurements relative to the variance of the phase-error are compared to those predicted by the linear PLL theory and the variance as computed from the approximate solutions. Finally, experimental results relative to the probability distribution of the time intervals between cycle-slipping events are given for signal-to-noise ratios in a range where the linear PLL theory does not apply. In particular, the maximum length of time the loop may be expected to remain in-lock is illustrated graphically as a function of signal-to-noise ratio in the loop bandwidth.

The main purpose of this paper was the transfer of the notion of regression into the theory of Banach valued random variables. This approach enables the formulation of identification problem in the domain of the infinite dimensional probability theory.

In Part I the Khintchine’s uniqueness theorem for the class convergence of probability distributions is proved in a natural way by making use of inverses of distribution functions; its generalization to the multidimensional case is also proved; relations between different paired sequences of scaling constants and centering constants in limit problems of probability distributions are given; and the general method to determine scaling constants and centering constants is presented

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.