Normal approximation for statistics of Gibbsian input in geometric probability

Abstract

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.