Convergence of random measures in geometric probability

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 Rd, we give a law of large numbers and central limit theorem for νn(f). The latter implies weak convergence of νn(·), suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications including the volume and surface measure of germ-grain models with unbounded grain sizes.