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Rates in almost sure invariance principle for slowly mixing dynamical systems. (English) Zbl 1448.37008

The authors prove the one-dimensional almost sure invariance principle with essentially optimal rates for polynomially mixing deterministic dynamical systems. A relevant example in this class is given by Pomeau-Manneville intermittent maps with Hölder continuous observables.
The novelty of the work lies in the representation of the dynamics as a Young tower-like Markov chain and in the application of the theory developed in [I. Berkes et al., Ann. Probab. 42, No. 2, 794–817 (2014; Zbl 1308.60037); C. Cuny et al., Stochastic Processes Appl. 128, No. 4, 1347–1385 (2018; Zbl 1384.60071); J. Komlós et al., Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111–131 (1975; Zbl 0308.60029); Z. Wahrscheinlichkeitstheor. Verw. Geb. 34, 33–58 (1976; Zbl 0307.60045)] for the approximation for dependent processes.

MSC:

37A50 Dynamical systems and their relations with probability theory and stochastic processes
37E05 Dynamical systems involving maps of the interval
60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
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References:

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