Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems

Ian Melbourne and Roland Zweimüller

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Abstract

We consider weak invariance principles (functional limit theorems) in the domain of a stable law. A general result is obtained on lifting such limit laws from an induced dynamical system to the original system. An important class of examples covered by our result are Pomeau–Manneville intermittency maps, where convergence for the induced system is in the standard Skorohod $\mathcal{J}_{1}$ topology. For the full system, convergence in the $\mathcal{J}_{1}$ topology fails, but we prove convergence in the $\mathcal{M}_{1}$ topology.

Résumé

Nous considérons des principes d’invariance faibles (théorèmes limites fonctionnels) dans le domaine d’une loi stable. Un résultat général est obtenu en relevant de telles lois limites depuis un système dynamique induit vers le système original. Une classe importante d’exemples couverte par notre résultat est donnée par les transformations intermittentes à la Pomeau–Manneville, où la convergence pour le système induit est dans la topologie $\mathcal{J}_{1}$ de Skorohod standard. Pour le système complet, il n’y a pas de convergence dans la topologie $\mathcal{J}_{1}$, mais nous prouvons la convergence dans la topologie $\mathcal{M}_{1}$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 2 (2015), 545-556.

Dates
First available in Project Euclid: 10 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1428672681

Digital Object Identifier
doi:10.1214/13-AIHP586

Mathematical Reviews number (MathSciNet)
MR3335015

Zentralblatt MATH identifier
1380.37064

Subjects
Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 28D05: Measure-preserving transformations 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 60F17: Functional limit theorems; invariance principles

Keywords
Nonuniformly hyperbolic systems Functional limit theorems Lévy processes Induced dynamical systems

Citation

Melbourne, Ian; Zweimüller, Roland. Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 2, 545--556. doi:10.1214/13-AIHP586. https://projecteuclid.org/euclid.aihp/1428672681


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