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.