CLT for random walks of commuting endomorphisms on compact abelian groups

Let $\text{Cal}S$ be an abelian group of automorphisms of a probability space $(X, {\Cal A}, \mu)$withafinitesystemofgenerators$(A_1, …, A_d)$\mathrm{.}Let$A^{\el}$denote$A_1^{\ell_1} … A_d^{\ell_d}$,for${\el}= (\ell_1, …, \ell_d)$\mathrm{.}If$(Z_k)$isarandomwalkon$\Z^d$, one can study the asymptotic distribution of the sums ${\sum }_{k=0}^{n-1}f\circ A^{Z_k(\omega )}$ and ${\sum }_{\text{el}\in {\mathbb{Z}}^d}\text{PP}(Z_n=\text{el})A^{\text{el}}f$, for a function $f$ on $X$. In particular, given a random walk on commuting matrices in $SL(\rho ,\mathbb{Z})$ or in ${\text{Cal}M}^{\ast }(\rho ,\mathbb{Z})$ acting on the torus ${\text{T}}^{\rho }$, $\rho \ge 1$, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on ${\text{T}}^{\rho }$ 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), $\text{Cal}S$ 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 $\text{Cal}S$, on random walks and on the variance of the associated ergodic sums.