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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

In a previous paper the author proved a distributional convergence for the Birkhoff sums of functions of null average defined over a dynamical system with an infinite, invariant, ergodic measure, akin to a central limit theorem. Here we extend this result to larger classes of observables, with milder smoothness conditions, and to larger classes of dynamical systems, which may no longer be mixing. A special emphasis is given to continuous time systems: semi-flows, flows, and Z d -extensions of flows. The latter generalization is applied to the geodesic flow on Z d -periodic manifolds of negative sectional curvature.

Z d -extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green-Kubo's formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve in turn the asumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on abelian covers of compact manifolds of negative curvature are discussed. arXiv:1702.06625v2 [math.DS] 16 May 2017 POTENTIAL KERNEL, HITTING PROBABILITIES AND DISTRIBUTIONAL ASYMPTOTICS 2 observables need only to decay polynomially at infinity, instead of having bounded support. We apply it to the geodesic flow on abelian covers of compact manifolds with negative curvature.This article is organized as follow. We present our setting and our results in Section 1, as well as our applications to Lorentz gases (Sub-subsection 1.4.1) and to geodesic flows (Sub-subsection 1.4.2). In Section 2 we present our spectral assumptions, and prove Theorem 1.4 using the method of moments. In Section 3 we prove Theorems 1.7 and 1.11, and in Section 4 the two applications mentioned above. We discuss Green-Kubo's formula in the Appendix.