A generalized central limit theorem in infinite ergodic theory

Abstract

International audienceWe prove a generalized central limit theorem for dynamical systems with an infinite ergodic measure which induce a Gibbs-Markov map on some subset, provided the return time to this subset has regularly varying tails. We adapt a method designed by Csaki and Foldes for observables of random walks to show that the partial sums of some functions of the system-the return time and the observable-are asymptotically independent. Some applications to random walks and Pomeau-Manneville maps are discussed

Variations on a central limit theorem in infinite ergodic theory

“…In [17], the author extended these results to discrete time dynamical systems which induce a Gibbs-Markov map on some subset. The method was adapted from the works of Csáki and Földes, and can be described as exploiting asymptotic independence via coupling methods.…”

“…These martingale methods allow us to easily extend our results to observables which take their values in a Hilbert space; this is done in §5. In §6, we replace the hypothesis of mixing which was made in [17] by ergodicity.…”

“…We begin by improving our previous results for discrete time systems. The critical points in the proof of the generalized CLT in [17] are a central limit theorem for the induced system, and a Burkholder-Rosenthal inequality. In §4, the smoothness condition used in [17] is relaxed.…”

“…The critical points in the proof of the generalized CLT in [17] are a central limit theorem for the induced system, and a Burkholder-Rosenthal inequality. In §4, the smoothness condition used in [17] is relaxed. As the new smoothness condition only guarantees a polynomial decay of correlations, spectral methods are no longer available, and we have to replace them by martingale methods.…”

“…We choose to approximate the height function by a discretized version, so as to apply the results of [17]. The main obstacle lies in extending the distributional convergence from one starting probability distribution to any absolutely continuous starting distribution, i.e.…”