Birkhoff sum convergence of Fr\'echet observables to stable laws for Gibbs-Markov systems and applications

TL;DR

This paper proves that Birkhoff sums of certain unbounded observables on Gibbs-Markov systems converge to stable laws, using a Poisson point process approach, with applications to intermittent maps.

Contribution

It introduces a Poisson point process method to establish stable law convergence for non-square-integrable observables on Gibbs-Markov systems, extending previous results to new settings.

Findings

Distributional convergence to stable laws for observables with heavy tails.

Verification of mixing conditions ensuring dominance of large values.

Application to intermittent-type maps demonstrating the interplay of effects.

Abstract

We use a Poisson point process approach to prove distributional convergence to a stable law for non square-integrable observables $\phi: [0,1]\to R$, mostly of the form $\phi (x) = d(x,x_0)^{-\frac{1}{\alpha}}$,$0<\alpha\le 2$, on Gibbs-Markov maps. A key result is to verify a standard mixing condition, which ensures that large values of the observable dominate the time-series, in the range $1<\alpha \le 2$. Stable limit laws for observables on dynamical systems have been established in two settings: ``good observables'' (typically H\"older) on slowly mixing non-uniformly hyperbolic systems and ``bad'' observables (unbounded with fat tails) on fast mixing dynamical systems. As an application we investigate the interplay between these two effects in a class of intermittent-type maps.