  The union of the particles of a stationary Poisson process of compact
(convex) sets in Euclidean space is called Boolean model and is a classical
topic of stochastic geometry. In this paper, Boolean models in hyperbolic space
are considered, where one takes the union of the particles of a stationary
Poisson process in the space of compact (convex) subsets of the hyperbolic
space. Geometric functionals such as the volume of the intersection of the
Boolean model with a compact convex observation window are studied. In
particular, the asymptotic behavior for balls with increasing radii as
observation windows is investigated. Exact and asymptotic formulas for
expectations, variances, and covariances are shown and univariate and
multivariate central limit theorems are derived. Compared to the Euclidean
framework, some new phenomena can be observed.
