Geometry of mathbb{Z}^d and the Central Limit Theorem for Weakly Dependent Random Fields

We study the asymptotic distribution of where A is a subset of , A_N= A[–N, N]^d, v(A) = lim_N card(A_N) (2N+1)^–d(0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, S_N(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if has a limit F(n; A) as N for each . We also study the class of sets A that satisfy this condition.