Central limit theorem and stable laws for intermittent maps

Abstract

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to onedimensional maps with a neutral fixed point at 0 of the form x + x 1+α , for α ∈ (0, 1). In particular, for α > 1/2, we show that the Birkhoff sums of a Hölder observable f converge to a normal law or a stable law, depending on whether f (0) = 0 or f (0) = 0. The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in non-commutative Banach algebras.

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

“…[RE83,GH88] for subshifts of finite type) when f ∈ L 2 . The article [Gou04] proves the same results under the slightly weaker assumption m(a)Df (a) < ∞. However, these methods are not sufficient to deal with the weaker assumption m(a)Df (a) η < ∞, hence new arguments will be required to prove the sufficiency part of Theorem 1.5.…”

Stable limits for sums of dependent infinite variance random variables

“…Both authors take pleasure in thanking their host institutions for excellent hospitality and financial support. We would like to thank the referee for bringing the paper by Gouëzel [29] to our attention. Remark 3 is also due to the referee.…”

Distributional and Local Limit Laws for a Class of Iterated Maps That Contract on Average

“…Stable limit laws have been well-understood for i.i.d. random variables [IL, for example], however there has been much recent interest in analogues of such results in dynamical systems, particularly in hyperbolic and non-uniformly hyperbolic systems [AD2,Gou, for example] and for random walks on the affine group of the real line [GP]. In this note we study iterated function schemes (IFS) with place-dependent probabilities that satisfy a 'contractionon-average' condition.…”