Concentration inequalities for measures of a Boolean model

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.