Rates in almost sure invariance principle for slowly mixing dynamical systems

Abstract

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau-Manneville intermittent maps, with Hölder continuous observables. Our rates have form $o(n^{\gamma }L(n))$, where $L(n)$ is a slowly varying function and $\gamma$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1/4})$. To break the $O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes-Liu-Wu and Cuny-Dedecker-Merlevède on the Komlós-Major-Tusnády approximation for dependent processes.

34 pages

References in corpus (4)

• A Borel-Cantelli lemma for intermittent interval maps
• Strong invariance principles with rate for "reverse" martingales and applications
• A note on stretched exponential decay of correlations for the Viana-Alves map
• On piecewise expanding maps

Cited by in corpus (1)

• Extensions with shrinking fibers