The Annals of Probability

A vector-valued almost sure invariance principle for hyperbolic dynamical systems

Ian Melbourne and Matthew Nicol

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We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Hölder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A diffeomorphisms and flows as well as systems modeled by Young towers with moderate tail decay rates.

In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a two-dimensional Brownian motion.

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Ann. Probab., Volume 37, Number 2 (2009), 478-505.

First available in Project Euclid: 30 April 2009

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Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D50: Hyperbolic systems with singularities (billiards, etc.) 60F17: Functional limit theorems; invariance principles

Almost sure invariance principle nonuniform hyperbolicity Lorentz gases Brownian motion Young towers


Melbourne, Ian; Nicol, Matthew. A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37 (2009), no. 2, 478--505. doi:10.1214/08-AOP410.

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