## Abstract

Let S be an abelian group of automorphisms of a probability space (X, A, μ) with a finite system of generators (A_{1}, … , 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.

Original language | English |
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Pages (from-to) | 143-195 |

Number of pages | 53 |

Journal | Journal of Theoretical Probability |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 2017 |

## Keywords

- Cumulant
- Mixing
- Quenched central limit theorem
- Random walk
- S-unit
- Self-intersections of a random walk
- Semigroup of endomorphisms
- Toral automorphism
- Z-action

## ASJC Scopus subject areas

- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty