Continuity and convergence of the percolation function in continuum percolation

1997 ◽  
Vol 34 (2) ◽  
pp. 363-371
Author(s):  
Anish Sarkar
Keyword(s):  
Euclidean Space ◽  
Critical Point ◽  
Poisson Point Process ◽  
Random Variable ◽  
Percolation Model ◽  
Dimensional Euclidean Space ◽  
Continuum Percolation ◽  
Positive Random Variable ◽  
Bounded Random Variable ◽  
Uniformly Bounded

We consider a percolation model on the d-dimensional Euclidean space which consists of spheres centred at the points of a Poisson point process of intensity ?. The radii of the spheres are random and are chosen independently and identically according to a distribution of a positive random variable. We show that the percolation function is continuous everywhere except perhaps at the critical point. Further, we show that the percolation functions converge to the appropriate percolation function except at the critical point when the radius random variables are uniformly bounded and converge weakly to another bounded random variable.

Continuity and convergence of the percolation function in continuum percolation

1997 ◽  
Vol 34 (02) ◽  
pp. 363-371
Author(s):  
Anish Sarkar
Keyword(s):  
Euclidean Space ◽  
Critical Point ◽  
Poisson Point Process ◽  
Random Variable ◽  
Percolation Model ◽  
Dimensional Euclidean Space ◽  
Continuum Percolation ◽  
Positive Random Variable ◽  
Bounded Random Variable ◽  
Uniformly Bounded

We consider a percolation model on the d-dimensional Euclidean space which consists of spheres centred at the points of a Poisson point process of intensity ?. The radii of the spheres are random and are chosen independently and identically according to a distribution of a positive random variable. We show that the percolation function is continuous everywhere except perhaps at the critical point. Further, we show that the percolation functions converge to the appropriate percolation function except at the critical point when the radius random variables are uniformly bounded and converge weakly to another bounded random variable.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

scholarly journals Recovering a Random Variable from Conditional Expectations Using Reconstruction Algorithms for the Gauss Radon Transform

2019 ◽  
pp. 1-31
Author(s):  
Jeremy Becnel ◽  
Daniel Riser-Espinoza
Keyword(s):  
Euclidean Space ◽  
Computerized Tomography ◽  
Radon Transform ◽  
Gaussian Measure ◽  
Random Variables ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Computer Implementation ◽  
Reconstruction Algorithms ◽  
Conditional Expectations

The Radon transform maps a function on n-dimensional Euclidean space onto its integral over a hyperplane. The fields of modern computerized tomography and medical imaging are fundamentally based on the Radon transform and the computer implementation of the inversion, or reconstruction, techniques of the Radon transform. In this work we use the Radon transform with a Gaussian measure to recover random variables from their conditional expectations. We derive reconstruction algorithms for random variables of unbounded support from samples of conditional expectations and discuss the error inherent in each algorithm.

scholarly journals Negatively dependent bounded random variable probability inequalities and the strong law of large numbers

2000 ◽  
Vol 13 (3) ◽  
pp. 261-267 ◽  
Author(s):  
M. Amini ◽  
A. Bozorgnia
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Random Variable ◽  
Strong Law ◽  
Probability Inequalities ◽  
Large Numbers ◽  
Bounded Random Variable ◽  
Strongly Consistent ◽  
Uniformly Bounded ◽  
Variable Probability

Let X1,…,Xn be negatively dependent uniformly bounded random variables with d.f. F(x). In this paper we obtain bounds for the probabilities P(|∑i=1nXi|≥nt) and P(|ξˆpn−ξp|>ϵ) where ξˆpn is the sample pth quantile and ξp is the pth quantile of F(x). Moreover, we show that ξˆpn is a strongly consistent estimator of ξp under mild restrictions on F(x) in the neighborhood of ξp. We also show that ξˆpn converges completely to ξp.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

Curve following in n—dimensional Euclidean space

SIMULATION ◽  
1973 ◽  
Vol 21 (5) ◽  
pp. 145-149 ◽  
Author(s):  
John Rees Jones
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space

scholarly journals On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

2014 ◽  
Vol 46 (3) ◽  
pp. 622-642 ◽  
Author(s):  
Julia Hörrmann ◽  
Daniel Hug
Keyword(s):  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Explicit Formula ◽  
Dimensional Euclidean Space ◽  
Arbitrary Dimensions ◽  
Zero Cell ◽  
Parametric Class

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to ∞.

Sign in / Sign up

Export Citation Format

Share Document