Central limit theorem for products of toral automorphisms

Jean-Pierre Conze (1) , Stéphane Le Borgne (1) , Mikaël Roger (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $(\tau_n)$ be a sequence of toral automorphisms $\tau_n : x \rightarrow A_n x \hbox{ mod }\ZZ^d$ with $A_n \in {\cal A}$, where ${\cal A}$ is a finite set of matrices in $SL(d, \mathbb{Z})$. Under some conditions the method of "multiplicative systems" of Komlòs can be used to prove a Central Limit Theorem for the sums $\sum_{k=1}^n f(\tau_k \circ \tau_{k-1} \cdots \circ \tau_1 x)$ if $f$ is a Hölder function on $\mathbb{T}^d$. These conditions hold for $2\times 2$ matrices with positive coefficients. In dimension $d$ they can be applied when $A_n= A_n(\omega)$, with independent choices of $A_n(\omega)$ in a finite set of matrices $\in SL(d, \mathbb{Z})$, in order to prove a "quenched" CLT.

Mots clés: Quenched central limit theorem; Random walk

Domaines: Probabilités [math.PR]

Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for products of toral automorphisms. 2010. ⟨hal-00491473v2⟩