CLT for random walks of commuting endomorphisms on compact abelian groups

Abstract

Let $\Cal S$ be an abelian group of automorphisms of a probability space $(X, {\Cal A}, μ)$ with a finite system of generators $(A_1,...,A_d)$. Let $A^{\el}$ denote $A_1^{{\ell }_1}...A_d^{{\ell }_d}$, for ${\el}= (\ell_1, ..., \ell_d)$. If $(Z_k)$ is a random walk on ${\mathbb{Z}}^d$, one can study the asymptotic distribution of the sums ${\sum }_{k=0}^{n-1}f\circ A^{Z_k(\omega )}$ and $\sum_{\el \in \Z^d} \PP(Z_n= \el) \, A^\el f$, for a function $f$ on $X$. In particular, given a random walk on commuting matrices in $SL(\rho ,\mathbb{Z})$ or in ${\Cal M}^*(ρ, \Z)$ acting on the torus $\T^ρ$, $\rho \ge 1$, 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), $\Cal S$ 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 $\Cal S$, on random walks and on the variance of the associated ergodic sums.