CLT for random walks of commuting endomorphisms on compact abelian groups

Let $\Cal S$ be an abelian group of automorphisms of a probability space $(X, {\Cal A}, \mu)$ with a finite system of generators $(A_1, ..., A_d)$. Let $A^{\el}$ denote $A_1^{\ell_1} ... A_d^{\ell_d}$, for ${\el}= (\ell_1, ..., \ell_d)$. If $(Z_k)$ is a random walk on $\Z^d$, one can study the asymptotic distribution of the sums $\sum_{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega)}}$ and $\sum_{\el \in \Z^d} \PP(Z_n= \el) \, A^\el f$, for a function $f$ on $X$. In particular, given a random walk on commuting matrices in $SL(\rho, \Z)$ or in ${\Cal M}^*(\rho, \Z)$ acting on the torus $\T^\rho$, $\rho \geq 1$, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on $\T^\rho$ after normalization? In this paper, we prove a central limit theorem when $X$ is a compact abelian connected group $G$ endowed with its Haar measure (e.g. a torus or a connected extension of a torus), $\Cal S$ a totally ergodic $d$-dimensional group of commuting algebraic automorphisms of $G$ and $f$ a regular function on $G$. The proof is based on the cumulant method and on preliminary results on the spectral properties of the action of $\Cal S$, on random walks and on the variance of the associated ergodic sums.