arXiv subjects

Zbyněk Pawlas

Publications and source records attributed to Zbyněk Pawlas.

5 recordsLinked to original sources

arXiv2026

Mixing Properties of Random Laguerre Tessellations

In this paper, we study the random Laguerre tessellation, a weighted generalization of the Voronoi tessellation, generated by a general stationary marked point process. We first derive a nearly optimal sufficient condition on the generating marked point process which ensures that the resulting random Laguerre tessellation is well-defined, utilising the concept of tempered configurations to handle potentially unbounded weights. We then investigate how the three mixing properties - ergodicity, mixing and $α$-mixing - of the generating marked point process are preserved for the corresponding random Laguerre tessellation. Our approach combines standard approximation arguments with the properties of tempered configurations and the measurability of the Laguerre mapping.

math.PR↗

arXiv2025

Random marked nested tessellations applied to the modelling of deformation twinning in polycrystalline materials

Stochastic geometry provides a powerful framework for modelling complex random structures, with applications in physics, materials science, biology, and other fields. The three-dimensional microstructure of polycrystalline materials is usually modeled by a randomly marked tessellation, where the marks correspond to crystallographic orientations. The purpose of this study is to extend the modelling approach to a finer scale, focusing on the subcells that emerge when a material specimen is exposed to mechanical loading. Specifically, the deformation twinning gives rise to nested tessellation, where the subcells are parallel twin lamellae and their complement is embedded within the original mother cells. The aim of this study is to develop a parametric mathematical model of marked nested tessellation and to realize it using stochastic simulations. We were able to deal with this model using computational tools. The sensitivity of the model to selected key parameters was investigated using statistical methods. As an application, a numerical simulation of the stress and strain fields resulting from deformation twinning is provided, and the contribution of the subcells to the total strain energy density under varying initial conditions was evaluated. This study highlights the dynamic capabilities of stochastic geometry in modelling a phenomenon that changes the microstructure.

math-ph↗

arXiv2023

Causality in extremes of time series

Consider two stationary time series with heavy-tailed marginal distributions. We aim to detect whether they have a causal relation, that is, if a change in one causes a change in the other. Usual methods for causal discovery are not well suited if the causal mechanisms only appear during extreme events. We propose a framework to detect a causal structure from the extremes of time series, providing a new tool to extract causal information from extreme events. We introduce the causal tail coefficient for time series, which can identify asymmetrical causal relations between extreme events under certain assumptions. This method can handle nonlinear relations and latent variables. Moreover, we mention how our method can help estimate a typical time difference between extreme events. Our methodology is especially well suited for large sample sizes, and we show the performance on the simulations. Finally, we apply our method to real-world space-weather and hydro-meteorological datasets.

math.ST↗

arXiv2019

Limit theory for unbiased and consistent estimators of statistics of random tessellations

We observe a realization of a stationary generalized weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling technique to construct an unbiased estimator of the average value of this geometric characteristic. Under mild conditions on the weights of the cells, we establish variance asymptotics and the asymptotic normality of the unbiased estimator as the observation window tends to the whole space. Moreover, the weak consistency is shown for this estimator.

math.PR↗

arXiv2013

Statistics for Poisson models of overlapping spheres

The paper considers the stationary Poisson Boolean model with spherical grains and proposes a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio-unbiased and asymptotically consistent for growing observation window. It is shown that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. The paper also provides a short discussion of related estimators as well as a simulation study.

math.PR↗