Rates in almost sure invariance principle for slowly mixing dynamical systems

2018/01/16 by Cuny, C., Dedecker, J., Korepanov, A. +1
#37E05 #60F17 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.05335

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^γL(n)), where L(n) is a slowly varying function and γ 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.

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