CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Guy Cohen, Jean Pierre Conze

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)143-195
Number of pages53
JournalJournal of Theoretical Probability
Volume30
Issue number1
DOIs
StatePublished - 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

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