February 2023 A functional stable limit theorem for Gibbs–Markov maps
David Kocheim, Fabian Pühringer, Roland Zweimüller
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 59(1): 185-207 (February 2023). DOI: 10.1214/22-AIHP1246


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 J1-topology on D([0,),Rd). The argument uses the classical approach via finite-dimensional marginals and J1-tightness. As applications, we record a Spitzer-type arcsine law for certain Z-extensions of Gibbs–Markov systems, and prove an asymptotic independence property of excursion processes of intermittent interval maps.

Pour une classe d’observables d-dimensionnelles localement (mais pas nécessairement uniformément) Lipschitz sur un système Gibbs–Markov, nous montrons que la convergence des sommes ergodiques (convenablement centrées et normalisées) vers un vecteur aléatoire de loi stable non gaussienne est équivalente au fait que la distribution appartient au domaine d’attraction classique. Dans ce cas nous montrons aussi un principe d’invariance faible dans la topologie forte de Skorohod J1 sur D([0,),Rd). L’argument utilise l’approche classique via les lois marginales de dimension finie et la tension de la suite dans le bon espace. Comme applications, nous démontrons une loi arcsinus à la Spitzer pour certaines Z-extensions des systèmes Gibbs–Markov, ainsi qu’une propriété d’indépendance asymptotique des processus d’excursion de certaines applications intermittentes de l’intervalle.


F. Pühringer gratefully acknowledges support through FWF-grant Y00782. R. Zweimüller thanks Jon Aaronson, Alexey Korepanov and Ian Melbourne for inspiring discussions related to this topic. We are also grateful to the referee whose comments helped to improve the presentation.


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David Kocheim. Fabian Pühringer. Roland Zweimüller. "A functional stable limit theorem for Gibbs–Markov maps." Ann. Inst. H. Poincaré Probab. Statist. 59 (1) 185 - 207, February 2023. https://doi.org/10.1214/22-AIHP1246


Received: 27 March 2020; Revised: 7 October 2021; Accepted: 10 January 2022; Published: February 2023
First available in Project Euclid: 16 January 2023

Digital Object Identifier: 10.1214/22-AIHP1246

Primary: 11K50 , 28D05 , 37A25 , 37C30

Keywords: Stable laws , Stable Lévy processes , Stationary sequences , Weak invariance principle , Weakly dependent processes

Rights: Copyright © 2023 Association des Publications de l’Institut Henri Poincaré


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Vol.59 • No. 1 • February 2023
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