Showing papers on "Random element published in 1978"
303 citations
TL;DR: The present paper points out that the entropy function h is a β -function, i.e., a monotone non-decreasing and submodular function with h(O) = 0 and that the pair ( E, h ) is a polymatroid.
Abstract: Given a finite set E of random variables, the entropy function h on E is a mapping from the set of all subsets of E into the set of all nonnegative real numbers such that for each A ⊆ E h(A) is the entropy of A . The present paper points out that the entropy function h is a β -function, i.e., a monotone non-decreasing and submodular function with h(O) = 0 and that the pair ( E, h ) is a polymatroid. The polymatroidal structure of a set of random variables induced by the entropy function is fundamental when we deal with the interdependence analysis of random variables such as the information-theoretic correlative analysis, the analysis of multiple-user communication networks, etc. Also, we introduce the notion of the principal partition of a set of random variables by transferring some results in the theory of matroids.
263 citations
TL;DR: In this paper, a method for analyzing flow through a porous medium whose parameters are random functions is presented, which is similar to the Galerkin formulation except the coefficients in the linear combination are no longer deterministic quantities but random functions.
Abstract: A method is presented for analyzing flow through a porous medium whose parameters are random functions. Such a medium is conceptualized as an ensemble of media with an associated probability mass function. The flow problem in each member of this ensemble is deterministic in the usual sense. All the solutions belong to a particular Hilbert space, and hence they can be written in terms of linear combinations of its basis functions. This is similar to the Galerkin formulation except that the coefficients in the linear combination are no longer deterministic quantities but random functions. The finite element method in conjunction with a Taylor series expansion is used to get the first two moments of the solution approximately. The method does not require specification of full probability mass functions of the parameters but only their first two moments, and spatial correlations can be easily accounted for. However, it is assumed that the probability mass functions are peaked at the expected value and are smooth in its vicinity. A sample problem is solved to illustrate the procedure. It is observed that the result is sensitive to the element size in the numerical scheme and the variances and spatial correlations of parameters. The expected value of the hydraulic head is found to differ significantly from the results that would have been obtained if the problem had been solved deterministically.
66 citations
TL;DR: In this article, it was shown that MP =M in distribution implies MP=M a.s.s for various classes of Markov chains, including aperiodic Harris recurrent chains, and that every Markov chain which has a random invariant measure also has a deterministic measure.
Abstract: Let P be the transition operator for a discrete time Markov chain on a space S. The object of the paper is to study the class of random measures on S which have the property that MP=M in distribution. These will be called random invariant measures for P. In particular, it is shown that MP=M in distribution implies MP=M a.s. for various classes of chains, including aperiodic Harris recurrent chains and aperiodic irreducible random walks. Some of this is done by exploiting the relationship between random invariant measures and entrance laws. These results are then applied to study the invariant probability measures for particle systems in which particles move independently in discrete time according to P. Finally, it is conjectured that every Markov chain which has a random invariant measure also has a deterministic invariant measure.
56 citations
TL;DR: In this article, it was shown that under certain conditions a continuous random operator with stochastic domain has a random fixed point provided that each realization has a (deterministic) fixed point.
41 citations
TL;DR: In this article, a random time change is defined as a map from one function space to another, and the continuity of this map is investigated, and applications are made to weak limit theorems of random processes.
40 citations
TL;DR: It is shown that a random process is spherically invariant if and only if it is equivalent to a zero-mean Gaussian process multiplied by an independent random variable.
Abstract: It is shown that a random process is spherically invariant if and only if it is equivalent to a zero-mean Gaussian process multiplied by an independent random variable. Several properties of spherically invariant random processes follow in a simple and direct fashion from this representation.
35 citations
TL;DR: In this paper, the convergence of weighted sums of tight random elements with zero expected values and uniformly bounded rth moments (r > 1) in a separable Banach space is obtained.
34 citations
31 citations
TL;DR: In this paper, a central limit theorem for Gibbs random fields satisfying a sufficiently strong mixing condition was proved for a finite subset of the lattice points of a fixed positive integer, where the random variables satisfy a mixing condition and partial sums have uniformly bounded variances.
Abstract: If $d$ is a fixed positive integer, let $\Lambda_N$ be a finite subset of $Z^d$, the lattice points of $\mathbb{R}^d$, with $\Lambda_N \uparrow Z^d$ and satisfying certain regularity properties. Let $(X_{N, Z})_{Z\in\Lambda_N}$ be a collection of random variables which satisfy a mixing condition and whose partial sums $X_N = \sum_{Z\in\Lambda_N} X_{N, Z}$ have uniformly bounded variances. Limit theorems, including a central limit theorem, are obtained for the sequence $X_N$. The results are applied to Gibbs random fields known to satisfy a sufficiently strong mixing condition.
TL;DR: The moments of the random proportion of a fixed set that is covered by a random set (moments of coverage) are shown to converge under very general conditions to the probability that the fixed set is almost everywhere covered by the random set as mentioned in this paper.
Abstract: The moments of the random proportion of a fixed set that is covered by a random set (moments of coverage) are shown to converge under very general conditions to the probability that the fixed set is almost everywhere covered by the random set. Moments and coverage probabilities are calculated for several cases of random arcs of random sizes on the circle. When comparing arc length distributions having the same expectation, it is conjectured that if one concentrates more mass near that expectation, the corresponding coverage probability will be smaller. Support for this conjecture is provided in special cases. GEOMETRICAL PROBABILITY; COVERAGE SPACE; MOMENTS OF COVERAGE;
TL;DR: In this paper, a central limit theorem with remainder and an iterated logarithm law for collections of mixing random variables indexed by Zd,d≧1 were proved for certain Gibbs random fields.
Abstract: We prove a central limit theorem with remainder and an iterated logarithm law for collections of mixing random variables indexed byZd,d≧1. These results are applicable to certain Gibbs random fields.
TL;DR: The exact distribution of fAnf, where An is a random determinant with independent and identically distributed exponential elements is given for the cases n = 2 and 3 in this article, and it is conjectured that for any fixed n, the probability density of [Anl for large values of the argument is the same as the density of (Y/n)n, where Y is a gamma random variable.
Abstract: The exact distribution of fAnf , where An is a random determinant with independent and identically distributed exponential elements is given for the cases n = 2 and 3 . From the investigation of the behaviour of the density functions for these cases it is conjectured that for any fixed n, the probability density of [Anl for large values of the argument is the same as the density of (Y/n)n , where Y is a gamma random variable.
CERN1
TL;DR: In this article, it was shown that the random voltage Vt resulting from pulses with independent random amplitude Yi Poisson arrivals, and exponential decay, can be asymptotically represented, in the stationary case, by the following random variable; namely a sum of products of random variables.
TL;DR: In this article, a derived random measure is constructed by integration of a random process with respect to a random measure independent of that process, and a central limit theorem for derived random measures is established.
TL;DR: In this article, the authors presented sharp bounds for the probabilities of the probability distributions of independent and identically distributed random variables under conditions on f and g. Results were obtained by an embedding method which approximates Brownian motion by sums of independent random variables taking on only two or three values.
Abstract: Let $S_1, S_2,\cdots$ be partial sums of independent and identically distributed random variables and let $f(n)$ and $g(n)$ be increasing positive sequences. Nearly sharp bounds are presented for the probabilities $P\{S_i \geqq g(i), i = 1,\cdots, n\}$ and $P\{- f(i) \leqq S_i \leqq f(i), i = 1,\cdots, n\}$ under conditions on $f$ and $g$. The most difficult results are the lower bounds in the normal case. Results are obtained by an embedding method which approximates Brownian motion by sums of independent random variables taking on only two or three values.
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TL;DR: In this paper, the authors introduce Levy random measures and present their basic properties, including the preservation of the Levy property under transformations of random measures (e.g., change of variable, passage to a limit) and under transformation of the probability laws of random measure is investigated.
Abstract: A Levy random measure is characterized by a conditional independence structure analogous to the Markov property. Here we introduce Levy random measures and present their basic properties. Preservation of the Levy property under transformations of random measures (e.g., change of variable, passage to a limit) and under transformations of the probability laws of random measures is investigated. One random measure is said to be a submeasure of a second random measure if its probability law is absolutely continuous with respect to that of the second. We show that if the second measure is a Levy random measure then the submeasure is Levy if and only if the Radon-Nikodym derivative satisfies a natural factorization condition. These results are applied to extend the theories of Gibbs states on bounded sets in $\mathbb{R}^
u$ and $\mathbf{Z}^
u$.
TL;DR: In this paper, the existence and uniqueness of a random solution of a non-linear perturbed random integral equation of the formx where teR+ and coωeΩ, the supporting set of a probability measure space, was studied.
Abstract: The object of the paper is concerned with the existence and uniqueness of a random solution, a second-order stochastic process, of a non-linear perturbed random integral equation of the formx where teR+ and coωeΩ, the supporting set of a probability measure space (Ω,A,μ). Several Jianach spaces and Bunnell's fixed point theorem are the primary techniques used.
TL;DR: In this article, partial sums of a non-degenerate sequence of Identically and independently distributed random variables taking values in a separable Hilbert space are shown to converge almost nowhere for 0 ≤ β ≤ 3/2.
Abstract: Let (Sn) be partial sums of a non-degenerate sequence of Identically and independently distributed random variables taking values in a separable Hilbert space. Then for 0 ≤ β ≤ 3/2, the series
converges almost nowhere. For β > 3/2 this may not be true.
TL;DR: The first characterizes the unit Gaussian random field by a strong independence property and the second determinesGaussian random fields that are generated by stochastic processes.
Abstract: Two results on Gaussian random fields are presented. The first characterizes the unit Gaussian random field by a strong independence property and the second determines Gaussian random fields that are generated by stochastic processes.
01 Jan 1978