CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let S be an abelian group of automorphisms of a probability space (X, A, μ) with a finite system of generators (A₁, … , A_d). Let Aℓ̲ denote A1ℓ1…Adℓd, for ℓ̲=(ℓ1,…,ℓd). If (Z_k) is a random walk on Z^d, one can study the asymptotic distribution of the sums ∑k=0n-1f∘AZk(ω) and ∑ℓ̲∈ZdP(Zn=ℓ̲)Aℓ̲f, for a function f on X. In particular, given a random walk on commuting matrices in SL(ρ, Z) or in M^∗(ρ, Z) acting on the torus T^ρ, ρ≥ 1 , what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on 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), 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.