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This paper concerns the asymptotic behavior of a random variable $W_\lambda$resulting from the summation of the functionals of a Gibbsian spatial pointprocess over windows $Q_\lambda \uparrow R^d$. We establish conditions ensuringthat $W_\lambda$ has volume order fluctuations, that is they coincide with thefluctuations of functionals of Poisson spatial point processes. We combine thisresult with Stein's method to deduce rates of normal approximation for$W_\lambda$, as $\lambda\to\infty$. Our general results establish varianceasymptotics and central limit theorems for statistics of random geometric andrelated Euclidean graphs on Gibbsian input. We also establish similar limittheory for claim sizes of insurance models with Gibbsian input, the number ofmaximal points of a Gibbsian sample, and the size of spatial birth-growthmodels with Gibbsian input.

Normal approximation of Gibbsian sums in geometric probability