Random Elds on Zz D , Limit Theorems and Irregular Sets

Consider a stationary random eld X = (X n) n2ZZ d and an additive functional of the form Z N (A; X) = X n2AN f N (X n) where A is an innnite subset of ZZ d , A N = A T ?N; N] d and f N is a real function, for instance f N (x) = (x?E(X0)) (2N +1) d=2 (normalized averages) or f N (x) = 1I fx>uNg for u N tending to innnity with N (high-level ex-ceedances). We will present results showing that if X is a weakly dependent random eld and A satisses certain geometrical conditions, limit theorems for Z n can be obtained. On the other hand, if A does not fulllll this conditions, then limit theorems can fail to hold, even for m-dependent X. We will also apply this results to derive limit theorems for additive functionals of some non-stationary models.