Rates of convergence in the multivariate weak invariance principle for nonuniformly hyperbolic maps

Abstract

We obtain rates of convergence in the weak invariance principle (functional central limit theorem) for ${\mathbb{R}}^d$-valued Hölder observables of nonuniformly hyperbolic maps. In particular, for maps modelled by a Young tower with superpolynomial tails (e.g. the Sinai billiard map, and Axiom A diffeomorphisms) we obtain a rate of $O(n^{-\kappa })$ in the Wasserstein $p$-metric for all $\kappa <1/4$ and $p<\infty$. Additionally, this is the first result on rates that covers certain invertible, slowly mixing maps, such as Bunimovich flowers.

arXiv comment14 pages, final version as appeared in Proc. AMS. Minor changes to the introduction

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