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Rates in almost sure invariance principle for slowly mixing dynamical systems

Part of:Low-dimensional dynamical systemsLimit theorems

Published online by Cambridge University Press:  26 February 2019

C. CUNY,
J. DEDECKER,
A. KOREPANOV and
F. MERLEVÈDE
C. CUNY
Affiliation:
Université de Brest, LMBA, UMR CNRS 6205, France email christophe.cuny@univ-brest.fr
J. DEDECKER
Affiliation:
Université Paris Descartes, Sorbonne Paris Cité, Laboratoire MAP5 (UMR 8145), France email jerome.dedecker@parisdescartes.fr
A. KOREPANOV
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK email a.korepanov@warwick.ac.uk
F. MERLEVÈDE
Affiliation:
Université Paris-Est, LAMA (UMR 8050), UPEM, CNRS, UPEC, France email florence.merlevede@u-pem.fr
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Abstract

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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 $o(n^{\unicode[STIX]{x1D6FE}}L(n))$, where $L(n)$ is a slowly varying function and $\unicode[STIX]{x1D6FE}$ 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.

Keywords

strong invariance principleKMT approximationnon-uniformly expanding dynamical systemsMarkov chain

MSC classification

Primary:37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary:60F17: Functional limit theorems; invariance principles

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020, pp. 2317 - 2348
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Copyright
© Cambridge University Press, 2019

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