NORMAL APPROXIMATION FOR STATISTICS OF GIBBSIAN INPUT IN GEOMETRIC PROBABILITY

This paper concerns the asymptotic behavior of a random variable W-lambda resulting from the summation of the functionals of a Gibbsian spatial point process over windows Q(lambda) up arrow R-d. We establish conditions ensuring that W-lambda 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-lambda as lambda -> infinity 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.