Convergence of random measures in geometric probability

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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.