A note on the almost sure central limit theorem for negatively associated fields

Let {X-i: i is an element of N-d} (d >= 1) be a field of negatively associated random variables. Set S-n = Sigma(i <= n) X-i, sigma(2)(n) = Var(S-n). Under some suitable conditions, we show that lim(N-->infinity) 1/D-N Sigma(k <= N)d(k) P (S-k/sigma(k) < x) = Phi(x) for any x is an element of R is a necessary and sufficient criteria for the almost sure central limit theorem, i.e. lim(N-->infinity) 1/D-N Sigma(k <= N)d(k) I (S-k/sigma(k) < x) = Phi(x) a.s. for any x is an element of R, where Phi(x) is the standard normal distribution function, D-N = Sigma(k <= N)d(k) and d(k) = 1/\k\exp(Sigma(d)(S=1)(log k(s))(alpha)), 0 <= alpha < 1/2, N is an element of N-d. In particular, we obtain here the almost sure central limit theorem for negatively associated fields that assures the usual central limit theorem. (C) 2008 Elsevier B.V. All rights reserved.