On the approximation of mean densities of random closed sets

TL;DR

This paper develops an approximation method for mean densities of regular random closed sets in , applicable to inhomogeneous processes, with examples illustrating its relevance to real-world phenomena.

Contribution

It introduces a general approximation framework for mean densities of regular random closed sets, extending known results and applicable to inhomogeneous spatial processes.

Findings

The approximation method is valid for sufficiently regular sets.

The approach generalizes existing results in the literature.

Examples demonstrate the method's applicability to various scenarios.

Abstract

Many real phenomena may be modelled as random closed sets in $\mathbb{R}^d$, of different Hausdorff dimensions. In many real applications, such as fiber processes and $n$-facets of random tessellations of dimension $n\leq d$ in spaces of dimension $d\geq1$, several problems are related to the estimation of such mean densities. In order to confront such problems in the general setting of spatially inhomogeneous processes, we suggest and analyze an approximation of mean densities for sufficiently regular random closed sets. We show how some known results in literature follow as particular cases. A series of examples throughout the paper are provided to illustrate various relevant situations.