Counter-example to the functional central limit theorem for real random fields

We consider the ergodic dynamical system (Omega, F, mu, T) with positive entropy where Omega is. a Lebesgue space, mu, is a probability measure and T is a measure preserving Z(d)-action, d is an element of N*. We show that, for any nonnegative real p, there is a real function f is an element of L-p(Omega) and a collection A of regular Borel sets of [0, 1](d) satisfying an entropy condition with inclusion such that (f o T-k)(kEZd) is a stationary martingale-difference random field but does not satisfy the functional central limit theorem (or invariance principle) with regard to the family A. (C) 2003 Editions scientifiques et medicales Elsevier SAS.