CLT for random walks of commuting endomorphisms on compact abelian groups

TL;DR

This paper establishes a central limit theorem for sums of functions along random walks generated by commuting automorphisms on compact abelian groups, extending classical results to a broader algebraic setting.

Contribution

It proves a CLT for ergodic sums along random walks on compact abelian groups with commuting automorphisms, using spectral and cumulant methods.

Findings

CLT holds for smooth functions on tori and connected groups under certain automorphism conditions.

Spectral properties of the automorphism group are crucial for the CLT proof.

Variance analysis supports the asymptotic normality of sums.

Abstract

Let $\Cal S$ be an abelian group of automorphisms of a probability space $(X, {\Cal A}, \mu)$ 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 $\Z^d$, 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 $SL(\rho, \Z)$ or in ${\Cal M}^*(\rho, \Z)$ acting on the torus $\T^\rho$, $\rho \geq 1$, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on $\T^\rho$ 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.