# Concentration inequalities for functionals of Poisson cylinder processes

## TL;DR

This paper develops concentration inequalities for the volume and intrinsic volumes of random union sets generated by Poisson processes of cylinders, extending classical results to more general geometric structures.

## Contribution

It introduces new concentration inequalities for functionals of Poisson cylinder processes, including cases with convex bases, isotropy, and expanding observation windows.

## Key findings

- Derived concentration inequalities for volume of Poisson cylinder union sets.
- Extended inequalities to intrinsic volumes under convexity and isotropy assumptions.
- Analyzed special cases including the classical Boolean model with zero-dimensional cylinders.

## Abstract

Random union sets $Z$ associated with stationary Poisson processes of $k$-cylinders in $\mathbb{R}^d$ are considered. Under general conditions on the typical cylinder base a concentration inequality for the volume of $Z$ restricted to a compact window is derived. Assuming convexity of the typical cylinder base and isotropy of $Z$ a concentration inequality for intrinsic volumes of arbitrary order is established. A number of special cases are discussed, for example the case when the cylinder bases arise from a random rotation of a fixed convex body. Also the situation of expanding windows is studied. Special attention is payed to the case $k=0$, which corresponds to the classical Boolean model.