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

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.