CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let $$\mathcal S$$ S be an abelian group of automorphisms of a probability space $$(X, {\mathcal A}, \mu )$$ ( X , A , μ ) with a finite system of generators $$(A_1, \ldots , A_d).$$ ( A 1 , … , A d ) . Let $$A^{{\underline{\ell }}}$$ A ℓ ̲ denote $$A_1^{\ell _1} \ldots A_d^{\ell _d}$$ A 1 ℓ 1 … A d ℓ d , for $${{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).$$ ℓ ̲ = ( ℓ 1 , … , ℓ d ) . If $$(Z_k)$$ ( Z k ) is a random walk on $${\mathbb {Z}}^d$$ Z d , one can study the asymptotic distribution of the sums $$\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}$$ ∑ k = 0 n - 1 f ∘ A Z k ( ω ) and $$\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f$$ ∑ ℓ ̲ ∈ Z d P ( Z n = ℓ ̲ ) A ℓ ̲ f , for a function f on X. In particular, given a random walk on commuting matrices in $$SL(\rho , {\mathbb {Z}})$$ S L ( ρ , Z ) or in $${\mathcal M}^*(\rho , {\mathbb {Z}})$$ M ∗ ( ρ , Z ) acting on the torus $${\mathbb {T}}^\rho $$ T ρ , $$\rho \ge 1$$ ρ ≥ 1 , what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on $${\mathbb {T}}^\rho $$ T ρ 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), $$\mathcal S$$ 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 random walks.