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\begin{document}$$\mathcal S$$\end{document} be an abelian group of automorphisms of a probability space (X,A,μ)\documentclass[12pt]{minimal}
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\begin{document}$$(X, {\mathcal A}, \mu )$$\end{document} with a finite system of generators (A1,…,Ad).\documentclass[12pt]{minimal}
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\begin{document}$$(A_1, \ldots , A_d).$$\end{document} Let Aℓ̲\documentclass[12pt]{minimal}
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\begin{document}$$A^{{\underline{\ell }}}$$\end{document} denote A1ℓ1…Adℓd\documentclass[12pt]{minimal}
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\begin{document}$$A_1^{\ell _1} \ldots A_d^{\ell _d}$$\end{document}, for ℓ̲=(ℓ1,…,ℓd).\documentclass[12pt]{minimal}
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\begin{document}$${{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).$$\end{document} If (Zk)\documentclass[12pt]{minimal}
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\begin{document}$$(Z_k)$$\end{document} is a random walk on Zd\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}^d$$\end{document}, one can study the asymptotic distribution of the sums ∑k=0n-1f∘AZk(ω)\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}$$\end{document} and ∑ℓ̲∈ZdP(Zn=ℓ̲)Aℓ̲f\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f$$\end{document}, for a function f on X. In particular, given a random walk on commuting matrices in SL(ρ,Z)\documentclass[12pt]{minimal}
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\begin{document}$$SL(\rho , {\mathbb {Z}})$$\end{document} or in M∗(ρ,Z)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal M}^*(\rho , {\mathbb {Z}})$$\end{document} acting on the torus Tρ\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}^\rho $$\end{document}, ρ≥1\documentclass[12pt]{minimal}
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\begin{document}$$\rho \ge 1$$\end{document}, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on Tρ\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}^\rho $$\end{document} 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), S\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal S$$\end{document} 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 random walks.
