Normal approximation for statistics of Gibbsian input in geometric probability

Abstract

This paper concerns the asymptotic behavior of a random variableWλresulting from the summation of the functionals of a Gibbsian spatial point process over windowsQλ↑ℝd. We establish conditions ensuring thatWλhas 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 forWλas 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.