A functional stable limit theorem for Gibbs-Markov maps

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.

arXiv commentSmall modifications meant to improve readability; 28 pages

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