A functional stable limit theorem for Gibbs-Markov maps

TL;DR

This paper establishes a functional stable limit theorem for Gibbs-Markov maps, showing convergence of normalized ergodic sums to non-Gaussian stable laws and demonstrating applications like arcsine laws and independence of excursion processes.

Contribution

It extends limit theorems to Gibbs-Markov systems with non-uniform Lipschitz observables, linking convergence to classical attraction domains and proving a weak invariance principle.

Findings

Convergence to non-Gaussian stable laws under certain conditions

A weak invariance principle in the Skorohod $ ext{J}_1$ topology

Applications include arcsine laws and independence of excursion processes

Abstract

For a class of locally (but not necessarily uniformly) Lipschitz continuous $d$-dimensional observables over a Gibbs-Markov system, we show that convergence of (suitably normalized and centered) ergodic sums to a non-Gaussian stable vector is equivalent to the distribution belonging to the classical domain of attraction, and that it implies a weak invariance principle in the (strong) Skorohod $\mathcal{J}_{1}$-topology on $\mathcal{D}([0,\infty),\mathbb{R}^{d})$. The argument uses the classical approach via finite-dimensional marginals and $\mathcal{J}_{1}$-tightness. As applications, we record a Spitzer-type arcsine law for certain $\mathbb{Z}% $-extensions of Gibbs-Markov systems, and prove an asymptotic independence property of excursion processes of intermittent interval maps.