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, ..., Ad). Let A l denote A1 1 ...A ld d , for l = (l1, ..., ld). If (Zk) is a random walk on Z , one can study the asymptotic distribution of the sums ∑n−1 k=0 f ◦AZk(ω) and ∑ l∈Z P(Zn = l)A f , for a function f on X . In particular, given a random walk on commuting matrices in SL(ρ,Z) or inM∗(ρ,Z) acting on the torus T, ρ ≥ 1, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on T 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 the spectral properties of the action of S, on random walks and on the variance of the associated ergodic sums.