# Concentration inequalities for measures of a Boolean model

## TL;DR

This paper derives concentration inequalities for measures of a Boolean model driven by a Poisson process, providing probabilistic bounds on the measure of the union of overlapping particles in a metric space.

## Contribution

It introduces new concentration inequalities for functionals of Poisson processes, specifically applied to measures of Boolean models with overlapping particles.

## Key findings

- Established concentration inequalities for
ho(Z) in Boolean models.
- Provided general Poisson process concentration inequalities applicable to various phase spaces.
- Enhanced understanding of probabilistic bounds for geometric random sets.

## Abstract

We consider a Boolean model $Z$ driven by a Poisson particle process $\eta$ on a metric space $\mathbb{Y}$. We study the random variable $\rho(Z)$, where $\rho$ is a (deterministic) measure on $\mathbb{Y}$. Due to the interaction of overlapping particles, the distribution of $\rho(Z)$ cannot be described explicitly. In this note we derive concentration inequalities for $\rho(Z)$. To this end we first prove two concentration inequalities for functions of a Poisson process on a general phase space.