Contre-exemple dans le théorème limite central pour les champs aléatoires réels

Mohamed El Machkouri (1) , Dalibor Volny (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

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. We show that for any nonnegative real $p$, there is a real function $f\in L^{p}(\Omega)$ and a collection $\A$ of regular Borel sets of $[0,1]^{d}$ satisfying an entropy condition with inclusion such that $(f\circ T^{k})_{k\in\Z^{d}}$ is a stationary martingale difference random field but does not satisfy the functional central limit theorem (or invariance principle) with regard to the family $\A$.

Mots clés: Martingale difference random field; Metric entropy; Functional central limit theorem; Invariance principle

Domaines: Probabilités [math.PR]

Mohamed El Machkouri, Dalibor Volny. Contre-exemple dans le théorème limite central pour les champs aléatoires réels. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2003, 39 (2), pp.325-337. ⟨hal-00488689⟩