This paper concerns the asymptotic behavior of a random variable Wλ resulting from the summation of the functionals of a Gibbsian spatial point process over windows Qλ ↑ ℝd. We establish conditions ensuring that Wλ 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 Wλ 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.


Functionals of spatial point process often satisfy a weak spatial dependencecondition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which isa level-3 result for empirical point fields as well as a level-2 resultfor empirical point measures. The level-3 rate function coincides withthe so-called specific information. We show that the general resultcan be applied to prove MDPs for various particular functionals,including random sequential packing, birth-growth models, germ-grainmodels and nearest neighbor graphs.