Geometric functionals of polyconvex excursion sets of Poisson shot noise processes

TL;DR

This paper investigates the geometric properties of excursion sets of specific Poisson shot noise processes, focusing on their asymptotic behavior, variance bounds, and distributional limits as observation windows grow large.

Contribution

It introduces a class of Poisson shot noise processes with polyconvex excursion sets and derives new asymptotic results, variance bounds, and a central limit theorem for their geometric functionals.

Findings

Asymptotic behavior of intrinsic volumes analyzed

Lower variance bounds established

Central limit theorem proved for geometric functionals

Abstract

Excursion sets of Poisson shot noise processes are a prominent class of random sets. We consider a specific class of Poisson shot noise processes whose excursion sets within compact convex observation windows are almost surely polyconvex. This class contains, for example, the Boolean model. In this paper, we analyse the behaviour of geometric functionals such as the intrinsic volumes of these excursion sets for growing observation windows. In particular, we study the asymptotics of the expectation and the variance, derive a lower variance bound and show a central limit theorem.