Gaussian limits for random geometric measures

Given n independent random marked d-vectors Xi with a common density, define the measure νn = ∑ i ξi, where ξi is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near Xi. Technically, this means here that ξi stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions f on R, we give a central limit theorem for νn(f), and deduce weak convergence of νn(·), suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications to measures associated with germ-grain models, random and cooperative sequential adsorption, Voronoi tessellation and k-nearest neighbours graph.