Almost sure behavior of functionals of stationary Gibbs point processes

This paper is concerned with the almost sure control of functionals of stationary Gibbs point processes. We apply Kahane–Khintchine’s inequality to derive an almost sure control of various functionals under very mild assumption on the spatial point process X. In particular, if X is a locally stable Gibbs point process with finite range observed in [−n,n]d, we obtain the bound N[−n,n]d(X)/(2n)d=ρ+Oa.s.(n−d/2logn3/2) as n→∞, where NW(X) is the number of points of X∩W for W⊂Rd and where ρ is the intensity parameter of  X.