Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

Abstract

Abstract. We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L 2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.

A Renewal Theorem for Strongly Ergodic Markov Chains in Dimension d ≥ 3 and Centered Case

“…As already observed in [9,Rk. 3.6], the passage to the m-differentiability properties of r z (·) can be deduced from a general lemma in [6].…”

“…Of course, the passage from (C') to the regularity properties of the maps (z − Q(·)) −1 is here more difficult than in the method (I) because (z − Q(t)) −1 must be seen as elements of L(B θ , B θ ) according to a procedure involving several suitable values (θ, θ ) ∈ I 2 . Such results have been presented in [9,10,15,17].…”

“…It can be also deduced from [9,Theorem 3.3] which specifies and generalizes the Taylor expansions obtained in [10,15]. Let us verify that the assumptions of [9,Theorem 3.3] are fulfilled. Define the spaces B m +1 := B ϑm ,…”

“…4 under the moment assumption ξ md ≤ C v. This is due to the fact that, when ξ is unbounded, the property Q ∈ C ε (R d , L(B θ )) is not fulfilled in general and must be replaced in the derivation procedure (of both [9,17] L(B θ , B θ )) with θ < θ . This yields a "gap" between the spaces B ϑm and B v (used in Lemma 6) which is slightly bigger than the expected one.…”

“…It is used in [11] to establish a one-dimensional (non-centered) Markov renewal theorem. Also mention that one of the more beautiful applications of this method concerns the convergence to stable laws, see [5,9,13].…”