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A note on the existence of Gibbs marked point processes with applications in stochastic geometry

Martina Petráková (2023)

Kybernetika

This paper generalizes a recent existence result for infinite-volume marked Gibbs point processes. We try to use the existence theorem for two models from stochastic geometry. First, we show the existence of Gibbs facet processes in d with repulsive interactions. We also prove that the finite-volume Gibbs facet processes with attractive interactions need not exist. Afterwards, we study Gibbs-Laguerre tessellations of 2 . The mentioned existence result cannot be used, since one of its assumptions...

A Simpler Proof of the Negative Association Property for Absolute Values of Measures Tied to Generalized Orlicz Balls

Jakub Onufry Wojtaszczyk (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Negative association for a family of random variables ( X i ) means that for any coordinatewise increasing functions f,g we have ( X i , . . . , X i k ) g ( X j , . . . , X j l ) f ( X i , . . . , X i k ) g ( X j , . . . , X j l ) for any disjoint sets of indices (iₘ), (jₙ). It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics by Alem Saxena and Joag-Dev Proschan, and brought to convex geometry in 2005 by Wojtaszczyk Pilipczuk to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple proof of...

Almost sure asymptotic behaviour of the r -neighbourhood surface area of Brownian paths

Ondřej Honzl, Jan Rataj (2012)

Czechoslovak Mathematical Journal

We show that whenever the q -dimensional Minkowski content of a subset A d exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in d , d 3 .

An approximation of the κ-function for the study of binary images.

Guillermo Ayala Gallego, Amelia Simó Vidal (1993)

Qüestiió

Jensen et al. (1990) gave an exact expression for the κ-function in non-overlapping Boolean models. The present study proposes and evaluates an approximate expression for the κ-function in overlapping isotropic Boolean models based on an approximation of the covariogram of the primary grain. We study the suitability of a Boolean model for two binary images using this approximate expression.

An incomplete Voronoi tessellation

Lutz Muche (1993)

Applicationes Mathematicae

This paper presents distributional properties of a random cell structure which results from a growth process. It starts at the points of a Poisson point process. The growth is spherical with identical speed for all points; it stops whenever the boundaries of different cells have contact. The whole process finally stops after time t. So the space is not completely filled with cells, and the cells have both planar and spherical boundaries. Expressions are given for contact distribution functions,...

Aproximación aleatoria de cuerpos convexos.

Fernando Affentranger (1992)

Publicacions Matemàtiques

Problems related to the random approximation of convex bodies fall into the field of integral geometry and geometric probabilities. The aim of this paper is to give a survey of known results about the stochastic model that has received special attention in the literature and that can be described as follows:Let K be a d-dimensional convex body in Eucliden space Rd, d ≥ 2. Denote by Hn the convex hull of n independent random points X1, ..., Xn distributed identically and uniformly in the interior...

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