Showing papers on "Random element published in 1984"

Chaotic motion and random matrix theories

Abstract: “Cada ciencia ha mester los vocables per los quals mills sia manifestada ; e car a aquesta c iencia ia demostrativa sien mester vocables s escurs e que los homens lecs no han en us, e cat nos facam aquest llibre als homens lecs per aco breument e ab plans vocables parlarem d' esta cienc cia” Ramon Llull (Libre de Gentil, 1273 ?)

Localization of random walks in one-dimensional random environments

Abstract: We consider a random walk on the one-dimensional semi-lattice ℤ={0, 1, 2,...}. We prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment. The size of this neighbourhood is estimated. The limit parameters of the walks are also determined.

Generating Correlation Matrices

Abstract: This paper describes a variety of methods for generating random correlation matrices, with emphasis on choice of random variables and distributions so as to provide matrices with given structure, expected values or eigenvalues.

Linear estimators and measurable linear transformations on a Hilbert space

Abstract: We consider the problem of estimating the mean of a Gaussian random vector with values in a Hilbert space. We argue that the natural class of linear estimators for the mean is the class of measurable linear transformations. We give a simple description of all measurable linear transformations with respect to a Gaussian measure. If X and θ are jointly Gaussian then E[θ¦X] is a measurable linear transformation. As an application of the general theory we describe all measurable linear transformations with respect to the Wiener measure in terms of Wiener integrals.

Recurrence and Transience Criteria for Random Walk in a Random Environment

Abstract: Oseledec's Multiplicative Ergodic Theorem is used to give recurrence and transience criteria for random walk in a random environment on the integers. These criteria generalize those given by Solomon in the nearest-neighbor case. The methodology for random environments is then applied to Markov chains with periodic transition functions to obtain recurrence and transience criteria for these processes as well.

Some random fixed point theorems for condensing operators

Abstract: In this paper we obtain several random fixed point theorems including a stochastic generalization of the classical Rothe fixed point theorem. The results herein improve a recent result of Bharucha-Reid and Mukherjea and also some similar results of Itoh. 1. Throughout, let (£2, 2) be a measurable space (2 = sigma algebra) and X a nonempty subset of a Banach space E. In a recent paper (1), Bharucha-Reid and Mukherjea (see also Mukherjea (6)) have given sufficient conditions for a stochastic analogue of Schauder's fixed point theorem for a random operator T : Q X X -» X. Itoh (5) introduced random condensing operators and considerably improved their result (see Lemma 1 below). In this paper, we consider random operators 7:0X1 -» E and give sufficient conditions for the existence of a measurable map :Q-+X satisfying a Browder-Fan (4) type result. As a consequence, a stochastic generaliza- tion of the well-known Rothe fixed point theorem is obtained. A random analogue of the Krasnoselskii fixed point theorem for the sum of two operators is given for a Hubert space.

Gram-de finetti matrices

Abstract: One investigates the probability structure of random sequences of elements of a Hilbert space, considered within the accuracy of an isometry, whose distributions are invariant relative to a finite permutation of the terms.

Convergence and Existence of Random Set Distributions

Abstract: We study the relation between distributions of random closed sets and their hitting functions $T$, defined by $T(B) = P\{\varphi \cap B eq \varnothing\}$ for Borel sets $B$. In particular, a sequence of random sets converges in distribution iff the corresponding sequence of hitting functions converges on some sufficiently large class of bounded Borel sets. This class may be chosen to be countable.

Parametric representation of mean values for stationary random mosaies

Abstract: Mean values referring to stationary random 2- or 3-dimensional, mosaics are considered, e.g. the mean number of vertices per unit volume, the mean content of the typical cell, the mean perimeter of the typical face, the mean number of emanating edges of the typical vertex etc, A parametric representation of some mean values is given in order to explain the relations between them, The parameters of the superposition of two independent isotropic random mosaics are evaluated

Persistent random walks in a one-dimensional random environment

Abstract: Central limit theorems are obtained for persistent random walks in a onedimensional random environment. They also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values.

Curvature Measures and Random Sets, I

Abstract: In choosing models of stochastic geometry three general problems play a role which are closely connected with each other: 1) Construction of the random geometric objects under consideration 2) Measurabilities 3) Geometric behaviour

Linear Congruential Generators Do Not Produce Random Sequences

Abstract: One of the most popular and fast methods of generating "random" sequence are linear congruential generators. This paper discusses the predictability of the sequence given only a constant proportion /spl alpha/ of the leading bits of the first few numbers generated. We show that the rest of the sequence is predictable in polynomial time, almost always, provided /spl alpha/ > 2/5.

On the Probability of Large Deviations in Banach Spaces

Abstract: Probabilities of large deviations for sums of i.i.d. Banach space valued random variables are investigated when the laws of the random variables converge weakly and a uniform exponential integrability condition is satisfied. Furthermore, a discussion of possible improvements of the estimates is given, when the probability is estimated that the sum lies in a convex set.

On the Cadlaguity of Random Measures

Abstract: We consider finitely additive random measures taking independent values on disjoint Borel sets in $R^k$, and ask when such measures, restricted to some subclass $\mathscr{A}$ of closed Borel sets, possess versions which are "right continuous with left limits", in an appropriate sense. The answer involves a delicate relationship between the "Levy measure" of the random measure and the size of $\mathscr{A}$, as measured via an entropy condition. Examples involving stable measures, Dudley's class $I(k, \alpha, M)$ of sets in $R^k$ with $\alpha$-times differentiable boundaries, and convex sets are considered as special cases, and an example given to show what can go wrong when the entropy of $\mathscr{A}$ is too large.

Noise-Induced Phases of Iterated Functions

Abstract: We consider iterating functions that consist of the identity map $[f(x)=x]$ plus a random function. The random function in every iteration is different and has a mean value of zero. We investigate the behavior of the entire iterated function. It is demonstrated that there are three distinct classes of random functions that generate three "phases" of the iterated function. These phases show universal properties independent of the precise form of the added random function. The physical interpretation of the model in terms of aggregation is discussed and an application of the above ideas is made to the problem of particles in a random potential that is varying in space and time.

Numerical solution for the problem of flame propagation by the random element method

Abstract: A numerical, grid-free algorithm is presented for the one-dimensional reaction-diffusion model of laminar flame propagation in premixed gases. It is based on the random element method developed by the authors for the analysis of diffusional processes. The effect of combustion is taken into account by applying the principle of fractional steps to separate the process of diffusion, modeled by the random walk of computational elements, from the exothermic effects of chemical reaction monitoring their strength. The validity of the algorithm is demonstrated by application to flame propagation problems for which exact solutions exist. The flame speed evaluated by its use oscillates around the exact value at a relatively small amplitude, while the temperature and species concentration profiles are self-correcting in their convergence to the exact solution. A satisfactory resolution is obtained by the use of a small number of computational elements which automatically adjust their distribution to fit sharp gradients.

An almost sure invariance principle for triangular arrays of banach space valued random variables

Abstract: We give a simpler proof of the probability invariance principle for triangular arrays of independent identically distributed random variables with values in a separable Banach space, recently proved by de Acosta [1], and improve this result to an almost sure invariance principle.

Stochastic convergence of randomly weighted sums of random elements

Abstract: Let {Xn} be random elements in a separable Banach space and let {ank} be an array of random variables. Convergence in probability and almost surely is obtained for the weighted sum under varying distributional and moment conditions on the random weights and on the random elements and geometric conditions on Banach spaces. In general, these results include the results for constant weights and real-valued random variables and are motivated in part by estimation problems and consistency considerations. Moreover, similar results are obtained for the space D[0, 1] under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements {Xn} on subintervals of a partition of [0,1] and represent significant improvements over existing laws o f large numbers and convergence results for weighted sums of random elements in D[0,1].

Factorization of probability measures and absolutely measurable sets

Abstract: We find necessary and sufficient conditions for a separable metric space Y to possess the property that for any measurable space (X, A) and probability measure P on X X Y, P can be factored. 1. Introduction. We characterize separable metric spaces Y which have the following property: for any measurable space (X, A) and any probability measure P on the product space (XxY,Ax B(Y)), where S(Z) will denote the Borel c-field of a metric space Z, P can be factored: P — QxT, where Q. is a probability measure on (X, A) and T: X x B(Y) —* (0,1) is an ^-measurable transition function such that ■