TY - JOUR

T1 - CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

AU - Cohen, Guy

AU - Conze, Jean Pierre

N1 - Funding Information:
This research was carried out during visits of the first author to the IRMAR at the University of Rennes 1 and of the second author to the Center for Advanced Studies in Mathematics at Ben Gurion University. The first author was partially supported by the ISF Grant 1/12. The authors are grateful to their hosts for their support. They thank Y. Guivarc’h, S. Le Borgne and M. Lin for helpful discussions and B. Weiss for the reference []. They thank also the referee for his/her careful reading and for the corrections and suggestions that greatly improved the presentation of the paper.
Publisher Copyright:
© 2015, European Union.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - Let S be an abelian group of automorphisms of a probability space (X, A, μ) with a finite system of generators (A1, … , Ad). Let Aℓ̲ denote A1ℓ1…Adℓd, for ℓ̲=(ℓ1,…,ℓd). If (Zk) is a random walk on Zd, 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 Tρ, ρ≥ 1 , what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on Tρ 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.

AB - Let S be an abelian group of automorphisms of a probability space (X, A, μ) with a finite system of generators (A1, … , Ad). Let Aℓ̲ denote A1ℓ1…Adℓd, for ℓ̲=(ℓ1,…,ℓd). If (Zk) is a random walk on Zd, 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 Tρ, ρ≥ 1 , what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on Tρ 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.

KW - Cumulant

KW - Mixing

KW - Quenched central limit theorem

KW - Random walk

KW - S-unit

KW - Self-intersections of a random walk

KW - Semigroup of endomorphisms

KW - Toral automorphism

KW - Z-action

UR - http://www.scopus.com/inward/record.url?scp=84939516101&partnerID=8YFLogxK

U2 - 10.1007/s10959-015-0631-y

DO - 10.1007/s10959-015-0631-y

M3 - Article

AN - SCOPUS:84939516101

SN - 0894-9840

VL - 30

SP - 143

EP - 195

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 1

ER -