This paper concerns the asymptotic behavior of a random variable W λ resulting from the summation of the functionals of a Gibbsian spatial point process over windows Q λ ↑ ℝ d . We establish conditions ensuring that W λ has volume order fluctuations, i.e. they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of a normal approximation for W λ as λ → ∞. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish a similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.

Let n points be placed uniformly at random in a subset A of the plane. A point is said to be maximal in the configuration if no other point is larger in both coordinates. We show that, for large n and for many sets A, the number of maximal points is approximately normally distributed. The argument uses Stein's method, and is also applicable in higher dimensions.