CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Abstract

Abstract. Let S be an abelian group of automorphisms of a probability space (X, A, µ) with a finite system of generators (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 the spectral properties of the action of S, on random walks and on the variance of the associated ergodic sums.

On the quenched functional CLT in random sceneries

“…For the action of commuting matrices in SL(ρ, Z) acting on T ρ , we prove for P-a.e. ω a functional CLT, extending previous results in [8]. A second algebraic example comes from commuting flows on homogeneous spaces.…”

“…There are C 0 , C finite positive constants 3 such that (cf. [5] Lemma 2.6, [20] Proposition 1.4 for ( 15) and ( 16), [8] Theorem 4.13 for ( 17)):…”

“…It can be shown that every summation associated to a random walk in Z d is ξ-regular for some measure ξ (cf. [8]).…”

“…First we recall some definitions and results about the random walks on Z d (see [26], details on the results recalled here can also be found in [8]).…”

A Functional CLT for Fields of Commuting Transformations Via Martingale Approximation

“…We see that if Ω 2,f (t) = o(| log(t)| −d/2−ε ) (t → 0) (25) holds. A CLT has been obtained by Cohen and Conze [2] under the condition Ω 2,f (t) = o(| log(t)| −d−ε ) (t → 0). However, the results of [2] apply to general arrays and to commuting families of general endomorphisms (including for instance automorphisms) inducing a totally ergodic N d -action (see their paper for more informations).…”