GEOMETRY OF Z(D) AND THE CENTRAL-LIMIT-THEOREM FOR WEAKLY DEPENDENT RANDOM-FIELDS

We study the asymptotic distribution of S-N(A, X) = root(2N + 1)(-d) ( Sigma(n is an element of AN) X-n), where A is a subset of Z(d), A(N) = A boolean AND[-N,N](d), v(A)=lim(N) card(A(N)) (2N + 1)(-d) is an ele ment of(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. M ore specifically, S-N(A, X) is asymptotically normal for each X that s atisfies certain mixting hypotheses if and only if F-N(n; A) = card{A( N)(c) boolean AND (n+A(N))}(2N + 1)(-d) has a limit F(n; A) as N --> i nfinity, for each n is an element of Z(d). We also study the class of sets A that satisfy this condition.