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1

Second-order theory for iteration stable tessellations

Schreiber T., Thäle C.

Probability and Mathematical Statistics

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2012

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Vol. 32, Fasc. 2

281--300

EN

This paper deals with iteration stable (STIT) tessellations, and, more generally, with a certain class of tessellations that are infinitely divisible with respect to iteration. They form a new, rich and flexible family of space-time models considered in stochastic geometry. The previously developed martingale tools are used to study second-order properties of STIT tessellations. A general formula for the variance of the total surface area of cell boundaries inside an observation window is shown. This general expression is combined with tools from integral geometry to derive exact and asymptotic second-order formulas in the stationary and isotropic regime. Also a general formula for the pair-correlation function of the surface measure is found.

2

Limit theory for planar Gilbert tessellations

Schreiber T., Soja N.

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 1

149--160

EN

A Gilbert tessellation arises by letting linear segments (cracks) in R2 unfold in time with constant speed, starting from a homogeneous Poisson point process of germs in randomly chosen directions. Whenever a growing edge hits an already existing one, it stops growing in this direction. The resulting process tessellates the plane. The purpose of the present paper is to establish a law of large numbers, variance asymptotics and a central limit theorem for geometric functionals of such tessellations. The main tool applied is the stabilization theory for geometric functionals.

3

Powstawanie kropli dla wielokątnych pól Markowa

Schreiber T.

Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne

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2008

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[Z] 44

15-22

EN

The paper discusses a class of continuum random contour models in the plane, going under the name of polygonal Markov fields as originally introduced by Arak & Surgailis and sharing a number of crucial features with the two-dimensional Ising model, which makes them interesting from the viewpoint of mathematical statistical physics. For such systems, modeling the co-existence of two opposing phases separated by polygonal contours, we present our results on low-temperature geometry of phase-separating interfaces. In this context, we show that in the phase transition regime the surplus of dominated phase creates a disk-shaped droplet surrounded by ocean of dominating phase (Wulff body) and minimising the model-specific surface energy functional. The proof is based on a particular graphical construction which also found its applications in digital image segmentation as indicated at the end of this article.

4

Association criteria for m-infinitely-divisible and u-infinitely-divisible random sets

Karłowska-Pik J., Schreiber T.

Probability and Mathematical Statistics

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2008

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Vol. 28, Fasc. 2

169--178

EN

We study random convex compact sets infinitely divisible with respect to the Minkowski addition and establish a sufficient condition for their association as well as a necessary and sufficient condition for the so-called infinite association. Further, we show also that every union infinitely-divisible random closed set and every convex compact set infinitely divisible for convex hulls of unions are associated.

5

Statistical inference from set-valued observations

Schreiber T.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 2

223--235

EN

Consider a random experiment whose true (unknown) outcome is modelled by a certain randomelementX and the available imprecise observations are modelled by some random set A such that X ∈ A almost surely. The purpose of the paper is to propose a statistical procedure for estimation of the real distribution of X. The asymptotic properties of the suggested procedure are then investigated in both nonparametric and parametric settings. So far, only the results for a finite sample space are available.

6

Large deviation principle for set-valued union processes

Schreiber T.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 2

273--285

EN

The purpose of the paper is to establish a large deviation principle for a certain class of increasing set-valued processes obeyingMarkovian dynamics. The obtained result is then applied to investigate the asymptotics of the sequence of successive convex hulls generated by uniform samples on a d-dimensional ball.