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

TL;DR

This work establishes explicit rates in the multivariate weak invariance principle for $\mathbb{R}^d$-valued Hölder observables of nonuniformly hyperbolic maps modeled by Young towers. By introducing the functional correlation bound and leveraging Bernstein’s big-block–small-block method, the author proves that $\mathcal{W}_r(W_n,W)\le C n^{-\kappa(\gamma,r)}$ for $1\le r<2\gamma$ with a concrete expression for $\kappa(\gamma,r)$, yielding rates approaching $1/4$ as $\gamma\to\infty$. The results apply to slowly mixing invertible systems, including Bunimovich flowers, and extend prior univariate rate results to the multivariate setting. The approach provides a unified pathway from decay of correlations to convergence rates in the functional CLT, with implications for statistical properties of a broad class of chaotic dynamics. The work thereby advances quantitative understanding of how nonuniform hyperbolicity and tail behavior influence rates in the functional central limit regime.

Abstract

We obtain rates of convergence in the weak invariance principle (functional central limit theorem) for $\R^d$-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 $O(n^{-κ})$ in the Wasserstein $p$-metric for all $κ<1/4$ and $p<\infty$. Additionally, this is the first result on rates that covers certain invertible, slowly mixing maps, such as Bunimovich flowers.

Theorems & Definitions (33)

• Example 1.1
• Example 1.2: Bunimovich flowers
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Theorem 2.4
• Remark 2.5
• Lemma 3.1
• proof
• Lemma 3.2