Showing papers on "Random element published in 1993"

Proper complex random processes with applications to information theory

Abstract: The covariance of complex random variables and processes, when defined consistently with the corresponding notion for real random variables, is shown to be determined by the usual complex covariance together with a quantity called the pseudo-covariance. A characterization of uncorrelatedness and wide-sense stationarity in terms of covariance and pseudo-covariance is given. Complex random variables and processes with a vanishing pseudo-covariance are called proper. It is shown that properness is preserved under affine transformations and that the complex-multivariate Gaussian density assumes a natural form only for proper random variables. The maximum-entropy theorem is generalized to the complex-multivariate case. The differential entropy of a complex random vector with a fixed correlation matrix is shown to be maximum if and only if the random vector is proper, Gaussian, and zero-mean. The notion of circular stationarity is introduced. For the class of proper complex random processes, a discrete Fourier transform correspondence is derived relating circular stationarity in the time domain to uncorrelatedness in the frequency domain. An application of the theory is presented. >

Small-bias probability spaces: efficient constructions and applications

Abstract: It is shown how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with “almost” equal probability. They are called $\epsilon $-biased random variables. The number of random bits needed to generate the random variables is $O(\log n + \log \frac{1}{\epsilon })$. Thus, if $\epsilon $ is polynomially small, then the size of the sample space is also polynomial. Random variables that are $\epsilon $-biased can be used to construct “almost” k-wise independent random variables where $\epsilon $ is a function of k.These probability spaces have various applications: l. Derandomization of algorithms: Many randomized algorithms that require only k-wise independence of their random bits (where k is bounded by $O(\log n)$), can be derandomized by using $\epsilon $-biased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive tes...

Probability, Random Variables and Random Signal Principles

Abstract: 1 Probability 2 The Random Variable 3 Operations on one Random Variable--Expectation 4 Multiple Random Variables 5 Operations of Multiple Random Variables 6 Random Processes-Temporal Characteristics 7 Random Processes-Spectral Characteristics 8 Linear Systems with Random Inputs 9 Optimum Linear Systems 10 Some Practical Applications of the Theory Appendix A Review of the Impulse Function Appendix B Gaussian Distribution Function Appendix C Useful Mathematical Quantities Appendix D Review of Fourier Transforms Appendix E Table of Useful Fourier Transforms Appendix F Some Probability Densities and Distributions Appendix G Some Mathematical Topics of Interest

Optimal discretization of random fields

Abstract: A new method for efficient discretization of random fields (i.e., their representation in terms of random variables) is introduced. The efficiency of the discretization is measured by the number of random variables required to represent the field with a specified level of accuracy. The method is based on principles of optimal linear estimation theory. It represents the field as a linear function of nodal random variables and a set of shape functions, which are determined by minimizing an error variance. Further efficiency is achieved by spectral decomposition of the nodal covariance matrix. The new method is found to be more efficient than other existing discretization methods, and more practical than a series expansion method employing the Karhunen‐Loeve theorem. The method is particularly useful for stochastic finite element studies involving random media, where there is a need to reduce the number of random variables so that the amount of required computations can be reduced.

Random Signals and Systems

Abstract: Introduction to random signals and probability a review of scalar random variables second order random vectors multidimensional random variables statistical description of random signals spectral properties of random signals statistical models for random signals poisson processes and affiliated signals random signals and dynamical systems mean square estimation estimation for stationary signals prediction for stationary signals time recursive methods matched filters.

Limit Theorems for Unions of Random Closed Sets

Abstract: Distributions of random closed sets.- Survey on stability of random sets and limit theorems for Minkowski addition.- Infinite divisibility and stability of random sets with respect to unions.- Limit theorems for normalized unions of random closed sets.- Almost sure convergence of unions of random closed sets.- Multivalued regularly varying functions and their applications to limit theorems for unions of random sets.- Probability metrics in the space of random sets distributions.- Applications of limit theorems.

On the weak coupling limit for a fermi gas in a random potential

Abstract: Under certain conditions on the initial state S of the Fermi gas and on the random potential V, the weak coupling limit (Van Hove limit) yields increase of entropy, propagation of chaos, convergence of the state for sufficiently small values of the rescaled time parameter r to a gauge-invariant and quasi-free asymptotic state, and the semigroup describing the evolution of the two-point function. Results are obtained not only for the average over the random potential but also with probability one.

Real symmetric random matrix ensembles of Hamiltonians with partial symmetry breaking.

Abstract: We investigate random matrix ensembles E(e) containing real symmetric matrices H=H (0) +eV, where H (0) is block diagonal, each block a member of the Gaussian orthogonal ensemble, coupled together by the Gaussian random elements of eV. E(e) could model, for example, a chaotic Hamiltonian apart from an approximate integral of the motion. We focus on transitions in eigenvalue and eigenvector projection statistics of E(e) upon variation of their respective scaling parameters. Expressions for the probability density of nearest-neighbor level spacings as well as the spectral rigidity are given, and supported by numerical data, and their application in determining a symmetry-breaking perturbation is discussed

Simulation of Random Functions on Large Lattices

Abstract: Random functions are in frequent use in applications of spatial statistics. Gaussian and Gaussian intrinsic random functions are differentiated, and the screening sequential algorithm for generating realizations from them are defined. The algorithm is based on the general sequential algorithm and Markov properties for random functions. For exponential and linear variogram functions the algorithm is shown to be exact for the one-dimensional case, while empirical evaluations show that it is highly reliable also in two-dimensional cases. For fractal random functions, the screening sequential algorithm is significally more reliable than the frequently used random midpoint displacement and successive random addition algorithms. The processing requirements for the algorithm is independent of the actual variogram function and linear in number of lattice nodes — both favorable characteristics.

Curve estimation in random fields of associated processes

Abstract: Consider a random field of real-valued positively or negatively associated random variables over the lattice of points in the d-dimensional Euclidean space with integer numbers as their coordinates, and suppose that the random variables are identically distributed with distribution function F and probability density function f. On the basis of an expanding segment of the underlying random field, the empirical distribution function is constructed, as well as kernel estimates for f, its derivatives, and the associated hazard rate. Under some additional weak conditions, it is shown that the empirical distribution function converges almost surely and uniformly to F, with rates. Similar results are established for the estimates of f, its derivatives, and the hazard rate, provided the support of F is a finite interval in the real line.

A method to generate autocorrelated uniform random numbers

Abstract: We present a simple, fast method to generate autocorrelated uniform random numbers. The “sum of uniforms” method adds a pair of U(0,1) random numbers, transforms the sum to a third U(0,1) random number, and uses this third random number as one member of the next pair. The method produces any desired level of positive or negative correlation between successive random numbers.

On deterministic and random fixed points

Abstract: Based on an extension of Aumann's measurable selection theorem due to Leese, it is shown that each fixed point theorem for F(ω, .) produces a random fixed point theorem for F provided the σ-algebra Σ for Ω is a Suslin family and F has a measurable graph (in particular, when F is random continuous with closed values and X is a separable metric space). As applications and illustrations, some random fixed points in the literature are obtained or extended

Random generalized set-valued quasi-complementarity problems

Abstract: The purpose of this paper is to introduce a class of random generalized set-valued quasi-complementarity problems and to construct a new random iterative algorithm. Some exis-tence theorems of random solutions to this kind of random quasi-complementarity problemsand some convergence theorems of random iterative sequences generated by this algorithmare shown. The results thus presented improve and extend some important recent results ofNoor.

On the Law of the Iterated Logarithm for Independent Banach Space Valued Random Variables

Abstract: In this paper we establish some general forms of the law of the iterated logarithm for independent random variables (Xn) with Banach space values, where (Xn) is not necessarily identically distributed. Our results include the Kolmogorov law of the iterated logarithm (LIL) in both finite and infinite dimensional cases, and they improve the Wittmann LIL as well as extend it to the vector setting. The Ledoux-Talagrand LIL for an i.i.d. sequence is also a simple corollary of our results.

Central limit theorem for finitely-dependent random variables

Abstract: For sequences of finitely-dependent random variables, under rather general hypotheses we establish estimates of integrals of the form where Fn(x) is the distribution function of the normalized sum of random variables; φ(x) is the standard normal distribution function. In the proof we use relations obtained by the method of C. Stein. The results are applicable, in particular, to m-dependent random variables and fields.

Quadratic Forms in Random Variables: Theory and Applications.

Abstract: Textbook for a one-semester graduate course for students specializing in mathematical statistics or in multivariate analysis, or reference for theoretical as well as applied statisticians, confines its discussion to quadratic forms and second degree polynomials in real normal random vectors and matr

Stationary measures for randomly chosen maps

Abstract: A Markov process on a compact metric space,X is given by random transformations.S is a finite set of continuous transformations ofX to itself. A random evolution onX is defined by lettingx inX evolve toT(x) forT inS with probability that depends onx andT but is independent of any other past measurable events. This type of model is often called a place dependent iterated function system. The transformations are assumed to have either monotone or contractive properties. Theorems are given to describe the number and types of ergodic invariant measures. Special emphasis is given to learning models and self-reinforcing random walks.

Probability and Random Processes - problems and solutions

Abstract: Events and their probabilities random variables and their distributions discrete random variables continuous random variables generating functions and their applications Markov chains convergence of random variables random processes stationary processes renewals queues Martingales diffusion processes.

Non-gaussian properties of second-order random waves

Abstract: Under the assumption that the second‐order random wave theory is valid, theoretical solutions for the probabilistic cumulants (or moments), in particular the third and fourth cumulants, of wave elevation associated with a deep‐water unidirectional random wave of arbitrary wave bandwidth are derived. In general, knowing the probability density function is not sufficient to obtain the corresponding power spectral density function, and vice versa. However, through the use of the second‐order random wave theory, the study shows that the probabilistic cumulants and spectral parameters of stationary random wave processes become closely related. In the present paper, numerical attention is given to random waves described by either Pierson‐Moskowitz, JONSWAP, or Wallops spectra. The paper also numerically verifies that the use of the second‐order random wave theory is appropriate only when the significant wave slope is less than about 0.02.

Random fixed points and approximations in random convex metric spaces

Abstract: ABSTCT Some random fixed point theorems in random convex metric spaces are obtained. Results regarding random best approximation on random convex metric spaces are also proved.